Determinants

In the first chapter we highlighted the special case of linear systems with the same number of equations as unknowns, those of the form $T \vec{x} = \vec{b}$ where $T$ is a square matrix. We noted that there are only two kinds of $T$ ’s. If $T$ is associated with a unique solution for any $\vec{b}$ , such as for the homogeneous system $T \vec{x} = \vec{0}$ , then $T$ is associated with a unique solution for every such $\vec{b}$ . We call such a matrix nonsingular. The other kind of $T$ , where every linear system for which it is the matrix of coefficients has either no solution or infinitely many solutions, we call singular.

In our work since then this distinction has been a theme. For instance, we now know that an $n \times n$ matrix $T$ is nonsingular if and only if each of these holds:

any system $T \vec{x} = \vec{b}$ has a solution and that solution is unique;
Gauss-Jordan reduction of $T$ yields an identity matrix;
the rows of $T$ form a linearly independent set;
the columns of $T$ form a linearly independent set, a basis for $ℝ^{n}$ ;
any map that $T$ represents is an isomorphism;
an inverse matrix $T^{- 1}$ exists.

So when we look at a square matrix, one of the first things that we ask is whether it is nonsingular.

This chapter develops a formula that determines whether $T$ is nonsingular. More precisely, we will develop a formula for $1 \times 1$ matrices, one for $2 \times 2$ matrices, etc. These are naturally related; that is, we will develop a family of formulas, a scheme that describes the formula for each size.

Since we will restrict the discussion to square matrices, in this chapter we will often simply say ‘matrix’ in place of ‘square matrix’.

Definition

Determining nonsingularity is trivial for $1 \times 1$ matrices.

$(\begin{matrix} a \end{matrix}) is nonsingular iff a \neq 0$

Corollary Three.IV.4.11 gives the $2 \times 2$ formula.

$(\begin{matrix} a & b \\ c & d \end{matrix}) is nonsingular iff a d - b c \neq 0$

We can produce the $3 \times 3$ formula as we did the prior one, although the computation is intricate (see Exercise 1.10).

$(\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}) is nonsingular iff a e i + b f g + c d h - h f a - i d b - g e c \neq 0$

With these cases in mind, we posit a family of formulas: $a$ , $a d - b c$ , etc. For each $n$ the formula defines a determinant function $\underset{n \times n}{det} : ℳ_{n \times n} \to ℝ$ such that an $n \times n$ matrix $T$ is nonsingular if and only if $\underset{n \times n}{det} (T) \neq 0$ . (We usually omit the subscript $n \times n$ because the size of $T$ describes which determinant function we mean.)

Exploration

This subsection is an optional motivation and development of the general definition. The definition is in the next subsection.

Above, in each case the matrix is nonsingular if and only if some formula is nonzero. But the three formulas don’t show an obvious pattern. We may spot that the $1 \times 1$ term $a$ has one letter, that the $2 \times 2$ terms $a d$ and $b c$ have two letters, and that the $3 \times 3$ terms each have three letters. We may even spot that in those terms there is a letter from each row and column of the matrix, e.g., in the $c d h$ term one letter comes from each row and from each column.

$(\begin{matrix} c \\ d \\ h \end{matrix})$

But these observations are perhaps more puzzling than enlightening. For instance, we might wonder why some terms are added but some are subtracted.

A good strategy for solving problems is to explore which properties the solution must have, and then search for something with those properties. So we shall start by asking what properties we’d like the determinant formulas to have.

At this point, our main way to decide whether a matrix is singular or not is to do Gaussian reduction and then check whether the diagonal of the echelon form matrix has any zeroes, that is, whether the product down the diagonal is zero. So we could guess that whatever determinant formula we find, the proof that it is right may involve applying Gauss’s Method to the matrix to show that in the end the product down the diagonal is zero if and only if our formula gives zero.

This suggests a plan: we will look for a family of determinant formulas that are unaffected by row operations and such that the determinant of an echelon form matrix is the product of its diagonal entries. In the rest of this subsection we will test this plan against the $2 \times 2$ and $3 \times 3$ formulas. In the end we will have to modify the “unaffected by row operations” part, but not by much.

First we check whether the $2 \times 2$ and $3 \times 3$ formulas are unaffected by the row operation of combining: if

$T ⟶_{\begin{array}{l} \end{array}}^{k ρ_{i} + ρ_{j}} \hat{T}$

then is $det (\hat{T}) = det (T)$ ? This check of the $2 \times 2$ determinant after the $k ρ_{1} + ρ_{2}$ operation

$det ((\begin{matrix} a & b \\ k a + c & k b + d \end{matrix})) = a (k b + d) - (k a + c) b = a d - b c$

shows that it is indeed unchanged, and the other $2 \times 2$ combination $k ρ_{2} + ρ_{1}$ gives the same result. Likewise, the $3 \times 3$ combination $k ρ_{3} + ρ_{2}$ leaves the determinant unchanged

$\begin{aligned} det ((\begin{array}{c} a & b & c \\ k g + d & k h + e & k i + f \\ g & h & i \end{array})) & = \begin{matrix} a (k h + e) i + b (k i + f) g + c (k g + d) h \\ - h (k i + f) a - i (k g + d) b - g (k h + e) c \end{matrix} \\ = a e i + b f g + c d h - h f a - i d b - g e c \end{aligned}$

as do the other $3 \times 3$ row combination operations.

So there seems to be promise in the plan. Of course, perhaps if we had worked out the $4 \times 4$ determinant formula and tested it then we might have found that it is affected by row combinations. This is an exploration and we do not yet have all the facts. Nonetheless, so far, so good.

Next we compare $det (\hat{T})$ with $det (T)$ for row swaps. Here we hit a snag: the $2 \times 2$ row swap $ρ_{1} \leftrightarrow ρ_{2}$ does not yield $a d - b c$ .

$det ((\begin{matrix} c & d \\ a & b \end{matrix})) = b c - a d$

And this $ρ_{1} \leftrightarrow ρ_{3}$ swap inside of a $3 \times 3$ matrix

$det ((\begin{matrix} g & h & i \\ d & e & f \\ a & b & c \end{matrix})) = g e c + h f a + i d b - b f g - c d h - a e i$

also does not give the same determinant as before the swap since again there is a sign change. Trying a different $3 \times 3$ swap $ρ_{1} \leftrightarrow ρ_{2}$

$det ((\begin{matrix} d & e & f \\ a & b & c \\ g & h & i \end{matrix})) = d b i + e c g + f a h - h c d - i a e - g b f$

also gives a change of sign.

So row swaps appear in this experiment to change the sign of a determinant. This does not wreck our plan entirely. We hope to decide nonsingularity by considering only whether the formula gives zero, not by considering its sign. Therefore, instead of expecting determinant formulas to be entirely unaffected by row operations we modify our plan so that on a swap they will change sign.

Obviously we finish by comparing $det (\hat{T})$ with $det (T)$ for the operation of multiplying a row by a scalar. This

$det ((\begin{matrix} a & b \\ k c & k d \end{matrix})) = a (k d) - (k c) b = k \cdot (a d - b c)$

ends with the entire determinant multiplied by $k$ , and the other $2 \times 2$ case has the same result. This $3 \times 3$ case ends the same way

$\begin{aligned} det ((\begin{array}{c} a & b & c \\ d & e & f \\ k g & k h & k i \end{array})) & = \begin{matrix} a e (k i) + b f (k g) + c d (k h) \\ - (k h) f a - (k i) d b - (k g) e c \end{matrix} \\ = k \cdot (a e i + b f g + c d h - h f a - i d b - g e c) \end{aligned}$

as do the other two $3 \times 3$ cases. These make us suspect that multiplying a row by $k$ multiplies the determinant by $k$ . As before, this modifies our plan but does not wreck it. We are asking only that the zero-ness of the determinant formula be unchanged, not focusing on the its sign or magnitude.

So in this exploration our plan got modified in some inessential ways and is now: we will look for $n \times n$ determinant functions that remain unchanged under the operation of row combination, that change sign on a row swap, that rescale on the rescaling of a row, and such that the determinant of an echelon form matrix is the product down the diagonal. In the next two subsections we will see that for each $n$ there is one and only one such function.

Finally, for the next subsection note that factoring out scalars is a row-wise operation: here

$det ((\begin{matrix} 3 & 3 & 9 \\ 2 & 1 & 1 \\ 5 & 11 & - 5 \end{matrix})) = 3 \cdot det ((\begin{matrix} 1 & 1 & 3 \\ 2 & 1 & 1 \\ 5 & 11 & - 5 \end{matrix}))$

the $3$ comes only out of the top row only, leaving the other rows unchanged. Consequently in the definition of determinant we will write it as a function of the rows $det ({\vec{ρ}}_{1}, {\vec{ρ}}_{2}, \dots {\vec{ρ}}_{n})$ , rather than as $det (T)$ or as a function of the entries $det (t_{1, 1}, \dots, t_{n, n})$ .

Exercises

Exercise 1.1 Supplied answer

Recommended. Evaluate the determinant of each.
1. $(\begin{matrix} 3 & 1 \\ - 1 & 1 \end{matrix})$
2. $(\begin{matrix} 2 & 0 & 1 \\ 3 & 1 & 1 \\ - 1 & 0 & 1 \end{matrix})$
3. $(\begin{matrix} 4 & 0 & 1 \\ 0 & 0 & 1 \\ 1 & 3 & - 1 \end{matrix})$
Back to Exercise 1.1

Answer.
1. $4$
2. $3$
3. $- 12$
Exercise 1.2 Supplied answer

Evaluate the determinant of each.
1. $(\begin{matrix} 2 & 0 \\ - 1 & 3 \end{matrix})$
2. $(\begin{matrix} 2 & 1 & 1 \\ 0 & 5 & - 2 \\ 1 & - 3 & 4 \end{matrix})$
3. $(\begin{matrix} 2 & 3 & 4 \\ 5 & 6 & 7 \\ 8 & 9 & 1 \end{matrix})$
Back to Exercise 1.2

Answer.
1. $6$
2. $21$
3. $27$
Exercise 1.3 Supplied answer

Recommended. Verify that the determinant of an upper-triangular $3 \times 3$ matrix is the product down the diagonal.

$det ((\begin{matrix} a & b & c \\ 0 & e & f \\ 0 & 0 & i \end{matrix})) = a e i$

Do lower-triangular matrices work the same way?

Back to Exercise 1.3

Answer. For the first, apply the formula in this section, note that any term with a $d$ , $g$ , or $h$ is zero, and simplify. Lower-triangular matrices work the same way.
Exercise 1.4 Supplied answer

Recommended. Use the determinant to decide if each is singular or nonsingular.
1. $(\begin{matrix} 2 & 1 \\ 3 & 1 \end{matrix})$
2. $(\begin{matrix} 0 & 1 \\ 1 & - 1 \end{matrix})$
3. $(\begin{matrix} 4 & 2 \\ 2 & 1 \end{matrix})$
Back to Exercise 1.4

Answer.
1. Nonsingular, the determinant is $- 1$ .
2. Nonsingular, the determinant is $- 1$ .
3. Singular, the determinant is $0$ .
Exercise 1.5 Supplied answer

Singular or nonsingular? Use the determinant to decide.
1. $(\begin{matrix} 2 & 1 & 1 \\ 3 & 2 & 2 \\ 0 & 1 & 4 \end{matrix})$
2. $(\begin{matrix} 1 & 0 & 1 \\ 2 & 1 & 1 \\ 4 & 1 & 3 \end{matrix})$
3. $(\begin{matrix} 2 & 1 & 0 \\ 3 & - 2 & 0 \\ 1 & 0 & 0 \end{matrix})$
Back to Exercise 1.5

Answer.
1. Nonsingular, the determinant is $3$ .
2. Singular, the determinant is $0$ .
3. Singular, the determinant is $0$ .
Exercise 1.6 Supplied answer

Recommended. Each pair of matrices differ by one row operation. Use this operation to compare $det (A)$ with $det (B)$ .
1. $A = (\begin{matrix} 1 & 2 \\ 2 & 3 \end{matrix})$ , $B = (\begin{matrix} 1 & 2 \\ 0 & - 1 \end{matrix})$
2. $A = (\begin{matrix} 3 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 2 \end{matrix})$ , $B = (\begin{matrix} 3 & 1 & 0 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{matrix})$
3. $A = (\begin{matrix} 1 & - 1 & 3 \\ 2 & 2 & - 6 \\ 1 & 0 & 4 \end{matrix})$ , $B = (\begin{matrix} 1 & - 1 & 3 \\ 1 & 1 & - 3 \\ 1 & 0 & 4 \end{matrix})$
Back to Exercise 1.6

Answer.
1. $det (B) = det (A)$ via $- 2 ρ_{1} + ρ_{2}$
2. $det (B) = - det (A)$ via $ρ_{2} \leftrightarrow ρ_{3}$
3. $det (B) = (1 / 2) \cdot det (A)$ via $(1 / 2) ρ_{2}$
Exercise 1.7 Supplied answer

Recommended. Find the determinant of this $4 \times 4$ matrix by following the plan: perform Gauss’s Method and look for the determinant to remain unchanged on a row combination, to change sign on a row swap, to rescale on the rescaling of a row, and such that the determinant of the echelon form matrix is the product down its diagonal.

$(\begin{matrix} 1 & 2 & 0 & 2 \\ 2 & 4 & 1 & 0 \\ 0 & 0 & - 1 & 3 \\ 3 & - 1 & 1 & 4 \end{matrix})$

Back to Exercise 1.7

Answer. Gauss’s Method does this.

$(\begin{matrix} 1 & 2 & 0 & 2 \\ 2 & 4 & 1 & 0 \\ 0 & 0 & - 1 & 3 \\ 3 & - 1 & 1 & 4 \end{matrix}) ⟶_{\begin{array}{l} - 3 ρ_{1} + ρ_{4} \end{array}}^{- 2 ρ_{1} + ρ_{2}} ⟶_{\begin{array}{l} \end{array}}^{ρ_{2} \leftrightarrow ρ_{4}} ⟶_{\begin{array}{l} \end{array}}^{ρ_{3} + ρ_{4}} (\begin{matrix} 1 & 2 & 0 & 2 \\ 0 & - 7 & 1 & - 2 \\ 0 & 0 & - 1 & 3 \\ 0 & 0 & 0 & - 1 \end{matrix})$

The echelon form matrix has a product down the diagonal of $1 \cdot (- 7) \cdot (- 1) \cdot (- 1) = - 7$ . In the course of Gauss’s Method no rows got rescaled but there was a row swap, so to get the determinant we change the sign, giving $+ 7$ .
Exercise 1.8 Supplied answer

Show this.

$det ((\begin{matrix} 1 & 1 & 1 \\ a & b & c \\ a^{2} & b^{2} & c^{2} \end{matrix})) = (b - a) (c - a) (c - b)$

Back to Exercise 1.8

Answer. Using the formula for the determinant of a $3 \times 3$ matrix we expand the left side

$1 \cdot b \cdot c^{2} + 1 \cdot c \cdot a^{2} + 1 \cdot a \cdot b^{2} - b^{2} \cdot c \cdot 1 - c^{2} \cdot a \cdot 1 - a^{2} \cdot b \cdot 1$

and by distributing we expand the right side.

$(b c - b a - a c + a^{2}) \cdot (c - b) = c^{2} b - b^{2} c - b a c + b^{2} a - a c^{2} + a c b + a^{2} c - a^{2} b$

Now we can just check that the two are equal. (Remark. This is the $3 \times 3$ case of Vandermonde’s determinant which arises in applications).
Exercise 1.9 Supplied answer

Recommended. Which real numbers $x$ make this matrix singular?

$(\begin{matrix} 12 - x & 4 \\ 8 & 8 - x \end{matrix})$

Back to Exercise 1.9

Answer. This equation

$0 = det ((\begin{matrix} 12 - x & 4 \\ 8 & 8 - x \end{matrix})) = 64 - 20 x + x^{2} = (x - 16) (x - 4)$

has roots $x = 16$ and $x = 4$ .
Exercise 1.10 Supplied answer

Do the Gaussian reduction to check the formula for $3 \times 3$ matrices stated in the preamble to this section.

$(\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix})$ is nonsingular iff $a e i + b f g + c d h - h f a - i d b - g e c \neq 0$

Back to Exercise 1.10

Answer. We first reduce the matrix to echelon form. To begin, assume that $a \neq 0$ and that $a e - b d \neq 0$ .

$\begin{matrix} ⟶_{\begin{array}{l} \end{array}}^{(1 / a) ρ_{1}} (\begin{matrix} 1 & b / a & c / a \\ d & e & f \\ g & h & i \end{matrix}) ⟶_{\begin{array}{l} - g ρ_{1} + ρ_{3} \end{array}}^{- d ρ_{1} + ρ_{2}} (\begin{matrix} 1 & b / a & c / a \\ 0 & (a e - b d) / a & (a f - c d) / a \\ 0 & (a h - b g) / a & (a i - c g) / a \end{matrix}) \\ ⟶_{\begin{array}{l} \end{array}}^{(a / (a e - b d)) ρ_{2}} (\begin{matrix} 1 & b / a & c / a \\ 0 & 1 & (a f - c d) / (a e - b d) \\ 0 & (a h - b g) / a & (a i - c g) / a \end{matrix}) \end{matrix}$

This step finishes the calculation.

$⟶_{\begin{array}{l} \end{array}}^{((a h - b g) / a) ρ_{2} + ρ_{3}} (\begin{matrix} 1 & b / a & c / a \\ 0 & 1 & (a f - c d) / (a e - b d) \\ 0 & 0 & (a e i + b g f + c d h - h f a - i d b - g e c) / (a e - b d) \end{matrix})$

Now assuming that $a \neq 0$ and $a e - b d \neq 0$ , the original matrix is nonsingular if and only if the $3, 3$ entry above is nonzero. That is, under the assumptions, the original matrix is nonsingular if and only if $a e i + b g f + c d h - h f a - i d b - g e c \neq 0$ , as required.

We finish by running down what happens if the assumptions that were taken for convenience in the prior paragraph do not hold. First, if $a \neq 0$ but $a e - b d = 0$ then we can swap

$(\begin{matrix} 1 & b / a & c / a \\ 0 & 0 & (a f - c d) / a \\ 0 & (a h - b g) / a & (a i - c g) / a \end{matrix}) ⟶_{\begin{array}{l} \end{array}}^{ρ_{2} \leftrightarrow ρ_{3}} (\begin{matrix} 1 & b / a & c / a \\ 0 & (a h - b g) / a & (a i - c g) / a \\ 0 & 0 & (a f - c d) / a \end{matrix})$

and conclude that the matrix is nonsingular if and only if either $a h - b g = 0$ or $a f - c d = 0$ . The condition ‘ $a h - b g = 0$ or $a f - c d = 0$ ’ is equivalent to the condition ‘ $(a h - b g) (a f - c d) = 0$ ’. Multiplying out and using the case assumption that $a e - b d = 0$ to substitute $a e$ for $b d$ gives this.

$\begin{matrix} 0 = a h a f - a h c d - b g a f + b g c d = a h a f - a h c d - b g a f + a e g c \\ = a (h a f - h c d - b g f + e g c) \end{matrix}$

Since $a \neq 0$ , we have that the matrix is nonsingular if and only if $h a f - h c d - b g f + e g c = 0$ . Therefore, in this $a \neq 0$ and $a e - b d = 0$ case, the matrix is nonsingular when $h a f - h c d - b g f + e g c - i (a e - b d) = 0$ .

The remaining cases are routine. Do the $a = 0$ but $d \neq 0$ case and the $a = 0$ and $d = 0$ but $g \neq 0$ case by first swapping rows and then going on as above. The $a = 0$ , $d = 0$ , and $g = 0$ case is easy—that matrix is singular since the columns form a linearly dependent set, and the determinant comes out to be zero.
Exercise 1.11 Supplied answer

Show that the equation of a line in $ℝ^{2}$ through $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is given by this determinant.

$det ((\begin{matrix} x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \end{matrix})) = 0 x_{1} \neq x_{2}$

Back to Exercise 1.11

Answer. Figuring the determinant and doing some algebra gives this.

$\begin{aligned} 0 & = y_{1} x + x_{2} y + x_{1} y_{2} - y_{2} x - x_{1} y - x_{2} y_{1} \\ (x_{2} - x_{1}) \cdot y & = (y_{2} - y_{1}) \cdot x + x_{2} y_{1} - x_{1} y_{2} \\ y & = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} \cdot x + \frac{x_{2} y_{1} - x_{1} y_{2}}{x_{2} - x_{1}} \end{aligned}$

Note that this is the equation of a line (in particular, in contains the familiar expression for the slope), and note that $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ satisfy it.
Exercise 1.12 Supplied answer

Many people have learned this mnemonic for the determinant of a $3 \times 3$ matrix: copy the first two columns to the right side of the matrix, then take the products down the forward diagonals and add them together, and then take the products on the backward diagonals and subtract them. That is, first write

$(\begin{array}{ccccc} h_{1, 1} & h_{1, 2} & h_{1, 3} & h_{1, 1} & h_{1, 2} \\ h_{2, 1} & h_{2, 2} & h_{2, 3} & h_{2, 1} & h_{2, 2} \\ h_{3, 1} & h_{3, 2} & h_{3, 3} & h_{3, 1} & h_{3, 2} \end{array})$

and then calculate this.

$\begin{array}{l} h_{1, 1} h_{2, 2} h_{3, 3} + h_{1, 2} h_{2, 3} h_{3, 1} + h_{1, 3} h_{2, 1} h_{3, 2} \\ - h_{3, 1} h_{2, 2} h_{1, 3} - h_{3, 2} h_{2, 3} h_{1, 1} - h_{3, 3} h_{2, 1} h_{1, 2} \end{array}$
1. Check that this agrees with the formula given in the preamble to this section.
2. Does it extend to other-sized determinants?
Back to Exercise 1.12

Answer.
1. The comparison with the formula given in the preamble to this section is easy.
2. While it holds for $2 \times 2$ matrices
  
  $\begin{aligned} (\begin{array}{ccc} h_{1, 1} & h_{1, 2} & h_{1, 1} \\ h_{2, 1} & h_{2, 2} & h_{2, 1} \end{array}) & = \begin{matrix} h_{1, 1} h_{2, 2} + h_{1, 2} h_{2, 1} \\ - h_{2, 1} h_{1, 2} - h_{2, 2} h_{1, 1} \end{matrix} \\ = h_{1, 1} h_{2, 2} - h_{1, 2} h_{2, 1} \end{aligned}$
  
  it does not hold for $4 \times 4$ matrices. An example is that this matrix is singular because the second and third rows are equal
  
  $(\begin{matrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ - 1 & 0 & 0 & 1 \end{matrix})$
  
  but following the scheme of the mnemonic does not give zero.
  
  $(\begin{array}{rrrrrrr} 1 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 & 1 & 1 \\ 0 & 1 & 1 & 0 & 0 & 1 & 1 \\ - 1 & 0 & 0 & 1 & - 1 & 0 & 0 \end{array}) = \begin{array}{l} 1 + 0 + 0 + 0 \\ - (- 1) - 0 - 0 - 0 \end{array}$
Exercise 1.13 Supplied answer

The cross product of the vectors

$\vec{x} = (\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}) \vec{y} = (\begin{matrix} y_{1} \\ y_{2} \\ y_{3} \end{matrix})$

is the vector computed as this determinant.

$\vec{x} \times \vec{y} = det ((\begin{matrix} {\vec{e}}_{1} & {\vec{e}}_{2} & {\vec{e}}_{3} \\ x_{1} & x_{2} & x_{3} \\ y_{1} & y_{2} & y_{3} \end{matrix}))$

Note that the first row’s entries are vectors, the vectors from the standard basis for $ℝ^{3}$ . Show that the cross product of two vectors is perpendicular to each vector.

Back to Exercise 1.13

Answer. The determinant is $(x_{2} y_{3} - x_{3} y_{2}) {\vec{e}}_{1} + (x_{3} y_{1} - x_{1} y_{3}) {\vec{e}}_{2} + (x_{1} y_{2} - x_{2} y_{1}) {\vec{e}}_{3}$ . To check perpendicularity, we check that the dot product with the first vector is zero

$(\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}) \cdot (\begin{matrix} x_{2} y_{3} - x_{3} y_{2} \\ x_{3} y_{1} - x_{1} y_{3} \\ x_{1} y_{2} - x_{2} y_{1} \end{matrix}) = x_{1} x_{2} y_{3} - x_{1} x_{3} y_{2} + x_{2} x_{3} y_{1} - x_{1} x_{2} y_{3} + x_{1} x_{3} y_{2} - x_{2} x_{3} y_{1} = 0$

and the dot product with the second vector is also zero.

$(\begin{matrix} y_{1} \\ y_{2} \\ y_{3} \end{matrix}) \cdot (\begin{matrix} x_{2} y_{3} - x_{3} y_{2} \\ x_{3} y_{1} - x_{1} y_{3} \\ x_{1} y_{2} - x_{2} y_{1} \end{matrix}) = x_{2} y_{1} y_{3} - x_{3} y_{1} y_{2} + x_{3} y_{1} y_{2} - x_{1} y_{2} y_{3} + x_{1} y_{2} y_{3} - x_{2} y_{1} y_{3} = 0$
Exercise 1.14 Supplied answer

Prove that each statement holds for $2 \times 2$ matrices.
1. The determinant of a product is the product of the determinants $det (S T) = det (S) \cdot det (T)$ .
2. If $T$ is invertible then the determinant of the inverse is the inverse of the determinant $det (T^{- 1}) = (det (T))^{- 1}$ .
Matrices $T$ and $T^{'}$ are similar if there is a nonsingular matrix $P$ such that $T^{'} = P T P^{- 1}$ . (We shall look at this relationship in Chapter Five.) Show that similar $2 \times 2$ matrices have the same determinant.
Back to Exercise 1.14

Answer.
1. Plug and chug: the determinant of the product is this
  
  $\begin{aligned} det ((\begin{array}{c} a & b \\ c & d \end{array}) (\begin{array}{c} w & x \\ y & z \end{array})) & = det ((\begin{array}{c} a w + b y & a x + b z \\ c w + d y & c x + d z \end{array})) \\ = \begin{matrix} a c w x + a d w z + b c x y + b d y z \\ - a c w x - b c w z - a d x y - b d y z \end{matrix} \end{aligned}$
  
  while the product of the determinants is this.
  
  $det ((\begin{matrix} a & b \\ c & d \end{matrix})) \cdot det ((\begin{matrix} w & x \\ y & z \end{matrix})) = (a d - b c) \cdot (w z - x y)$
  
  Verification that they are equal is easy.
2. Use the prior item.
That similar matrices have the same determinant is immediate from the above two: $det (P T P^{- 1}) = det (P) \cdot det (T) \cdot det (P^{- 1})$ .
Exercise 1.15 Supplied answer

Recommended. Prove that the area of this region in the plane

is equal to the value of this determinant.

$det ((\begin{matrix} x_{1} & x_{2} \\ y_{1} & y_{2} \end{matrix}))$

Compare with this.

$det ((\begin{matrix} x_{2} & x_{1} \\ y_{2} & y_{1} \end{matrix}))$

Back to Exercise 1.15

Answer. One way is to count these areas

by taking the area of the entire rectangle and subtracting the area of $A$ the upper-left rectangle, $B$ the upper-middle triangle, $D$ the upper-right triangle, $C$ the lower-left triangle, $E$ the lower-middle triangle, and $F$ the lower-right rectangle $(x_{1} + x_{2}) (y_{1} + y_{2}) - x_{2} y_{1} - (1 / 2) x_{1} y_{1} - (1 / 2) x_{2} y_{2} - (1 / 2) x_{2} y_{2} - (1 / 2) x_{1} y_{1} - x_{2} y_{1}$ . Simplification gives the determinant formula.

This determinant is the negative of the one above; the formula distinguishes whether the second column is counterclockwise from the first.
Exercise 1.16 Supplied answer

Prove that for $2 \times 2$ matrices, the determinant of a matrix equals the determinant of its transpose. Does that also hold for $3 \times 3$ matrices?

Back to Exercise 1.16

Answer. The computation for $2 \times 2$ matrices, using the formula quoted in the preamble, is easy. It does also hold for $3 \times 3$ matrices; the computation is routine.
Exercise 1.17 Supplied answer

Is the determinant function linear —is $det (x \cdot T + y \cdot S) = x \cdot det (T) + y \cdot det (S)$ ?

Back to Exercise 1.17

Answer. No. We illustrate with the $2 \times 2$ determinant. Recall that constants come out one row at a time.

$det ((\begin{matrix} 2 & 4 \\ 2 & 6 \end{matrix})) = 2 \cdot det ((\begin{matrix} 1 & 2 \\ 2 & 6 \end{matrix})) = 2 \cdot 2 \cdot det ((\begin{matrix} 1 & 2 \\ 1 & 3 \end{matrix}))$

This contradicts linearity (here we didn’t need $S$ , i.e., we can take $S$ to be the matrix of zeros).
Exercise 1.18 Supplied answer

Show that if $A$ is $3 \times 3$ then $det (c \cdot A) = c^{3} \cdot det (A)$ for any scalar $c$ .

Back to Exercise 1.18

Answer. Bring out the $c$ ’s one row at a time.
Exercise 1.19 Supplied answer

Which real numbers $θ$ make

$(\begin{matrix} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{matrix})$

singular? Explain geometrically.

Back to Exercise 1.19

Answer. There are no real numbers $θ$ that make the matrix singular because the determinant of the matrix $\cos^{2} θ + \sin^{2} θ$ is never $0$ , it equals $1$ for all $θ$ . Geometrically, with respect to the standard basis, this matrix represents a rotation of the plane through an angle of $θ$ . Each such map is one-to-one —for one thing, it is invertible.
Exercise 1.20 Supplied answer

Puzzle. [Am. Math. Mon., Apr. 1955] If a third order determinant has elements $1$ , $2$ , …, $9$ , what is the maximum value it may have?

Back to Exercise 1.20

Answer. This is how the answer was given in the cited source. Let $P$ be the sum of the three positive terms of the determinant and $- N$ the sum of the three negative terms. The maximum value of $P$ is

$9 \cdot 8 \cdot 7 + 6 \cdot 5 \cdot 4 + 3 \cdot 2 \cdot 1 = 630.$

The minimum value of $N$ consistent with $P$ is

$9 \cdot 6 \cdot 1 + 8 \cdot 5 \cdot 2 + 7 \cdot 4 \cdot 3 = 218.$

Any change in $P$ would result in lowering that sum by more than $4$ . Therefore $412$ the maximum value for the determinant and one form for the determinant is

$| \begin{matrix} 9 & 4 & 2 \\ 3 & 8 & 6 \\ 5 & 1 & 7 \end{matrix} | .$

Properties of Determinants

We want a formula to determine whether an $n \times n$ matrix is nonsingular. We will not begin by stating such a formula. Instead we will begin by considering, for each $n$ , the function that such a formula calculates. We will define this function by a list of properties. We will then prove that a function with these properties exists and is unique, and also describe how to compute it. (Because we will eventually prove this, from the start we will just say ‘ $det (T)$ ’ instead of ‘if there is a unique determinant function then $det (T)$ ’.)

Definition 2.1 A $n \times n$ determinant is a function $det : ℳ_{n \times n} \to ℝ$ such that

$det ({\vec{ρ}}_{1}, \dots, k \cdot {\vec{ρ}}_{i} + {\vec{ρ}}_{j}, \dots, {\vec{ρ}}_{n}) = det ({\vec{ρ}}_{1}, \dots, {\vec{ρ}}_{j}, \dots, {\vec{ρ}}_{n})$ for $i \neq j$
$det ({\vec{ρ}}_{1}, \dots, {\vec{ρ}}_{j}, \dots, {\vec{ρ}}_{i}, \dots, {\vec{ρ}}_{n}) = - det ({\vec{ρ}}_{1}, \dots, {\vec{ρ}}_{i}, \dots, {\vec{ρ}}_{j}, \dots, {\vec{ρ}}_{n})$ for $i \neq j$
$det ({\vec{ρ}}_{1}, \dots, k {\vec{ρ}}_{i}, \dots, {\vec{ρ}}_{n}) = k \cdot det ({\vec{ρ}}_{1}, \dots, {\vec{ρ}}_{i}, \dots, {\vec{ρ}}_{n})$ for any scalar $k$
$det (I) = 1$ where $I$ is an identity matrix

(the $\vec{ρ}$ ’s are the rows of the matrix). We often write $| T |$ for $det (T)$ .

Remark 2.2 Condition (2) is redundant since

$T ⟶_{\begin{array}{l} \end{array}}^{ρ_{i} + ρ_{j}} ⟶_{\begin{array}{l} \end{array}}^{- ρ_{j} + ρ_{i}} ⟶_{\begin{array}{l} \end{array}}^{ρ_{i} + ρ_{j}} ⟶_{\begin{array}{l} \end{array}}^{- ρ_{i}} \hat{T}$

swaps rows $i$ and $j$ . We have listed it for consistency with the Gauss’s Method presentation in earlier chapters.

Remark 2.3 Condition (3) does not have a $k \neq 0$ restriction, although the Gauss’s Method operation of multiplying a row by $k$ does have it. The next result shows that we do not need that restriction here.

Lemma 2.4 A matrix with two identical rows has a determinant of zero. A matrix with a zero row has a determinant of zero. A matrix is nonsingular if and only if its determinant is nonzero. The determinant of an echelon form matrix is the product down its diagonal.

Proof To verify the first sentence swap the two equal rows. The sign of the determinant changes but the matrix is the same and so its determinant is the same. Thus the determinant is zero.

For the second sentence multiply the zero row by two. That doubles the determinant but it also leaves the row unchanged, and hence leaves the determinant unchanged. Thus the determinant must be zero.

Do Gauss-Jordan reduction for the third sentence, $T \to \dots \to \hat{T}$ . By the first three properties the determinant of $T$ is zero if and only if the determinant of $\hat{T}$ is zero (although the two could differ in sign or magnitude). A nonsingular matrix $T$ Gauss-Jordan reduces to an identity matrix and so has a nonzero determinant. A singular $T$ reduces to a $\hat{T}$ with a zero row; by the second sentence of this lemma its determinant is zero.

The fourth sentence has two cases. If the echelon form matrix is singular then it has a zero row. Thus it has a zero on its diagonal and the product down its diagonal is zero. By the third sentence of this result the determinant is zero and therefore this matrix’s determinant equals the product down its diagonal.

If the echelon form matrix is nonsingular then none of its diagonal entries is zero. This means that we can divide by those entries and use condition (3) to get $1$ ’s on the diagonal.

$| \begin{matrix} t_{1, 1} & t_{1, 2} & t_{1, n} \\ 0 & t_{2, 2} & t_{2, n} \\ ⋱ \\ 0 & t_{n, n} \end{matrix} | = t_{1, 1} \cdot t_{2, 2} \dots t_{n, n} \cdot | \begin{matrix} 1 & t_{1, 2} / t_{1, 1} & t_{1, n} / t_{1, 1} \\ 0 & 1 & t_{2, n} / t_{2, 2} \\ ⋱ \\ 0 & 1 \end{matrix} |$

Then the Jordan half of Gauss-Jordan elimination leaves the identity matrix.

$= t_{1, 1} \cdot t_{2, 2} \dots t_{n, n} \cdot | \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ ⋱ \\ 0 & 1 \end{matrix} | = t_{1, 1} \cdot t_{2, 2} \dots t_{n, n} \cdot 1$

So in this case also, the determinant is the product down the diagonal.

QED

That gives us a way to compute the value of a determinant function on a matrix: do Gaussian reduction, keeping track of any changes of sign caused by row swaps and any scalars that we factor out, and finish by multiplying down the diagonal of the echelon form result. This algorithm is as fast as Gauss’s Method and so is practical on all of the matrices that we will see.

Example 2.5 Doing $2 \times 2$ determinants with Gauss’s Method

$| \begin{matrix} 2 & 4 \\ - 1 & 3 \end{matrix} | = | \begin{matrix} 2 & 4 \\ 0 & 5 \end{matrix} | = 10$

doesn’t give a big time savings because the $2 \times 2$ determinant formula is easy. However, a $3 \times 3$ determinant is often easier to calculate with Gauss’s Method than with its formula.

$| \begin{matrix} 2 & 2 & 6 \\ 4 & 4 & 3 \\ 0 & - 3 & 5 \end{matrix} | = | \begin{matrix} 2 & 2 & 6 \\ 0 & 0 & - 9 \\ 0 & - 3 & 5 \end{matrix} | = - | \begin{matrix} 2 & 2 & 6 \\ 0 & - 3 & 5 \\ 0 & 0 & - 9 \end{matrix} | = - 54$

Example 2.6 Determinants bigger than $3 \times 3$ go quickly with the Gauss’s Method procedure.

$| \begin{matrix} 1 & 0 & 1 & 3 \\ 0 & 1 & 1 & 4 \\ 0 & 0 & 0 & 5 \\ 0 & 1 & 0 & 1 \end{matrix} | = | \begin{matrix} 1 & 0 & 1 & 3 \\ 0 & 1 & 1 & 4 \\ 0 & 0 & 0 & 5 \\ 0 & 0 & - 1 & - 3 \end{matrix} | = - | \begin{matrix} 1 & 0 & 1 & 3 \\ 0 & 1 & 1 & 4 \\ 0 & 0 & - 1 & - 3 \\ 0 & 0 & 0 & 5 \end{matrix} | = - (- 5) = 5$

That example raises an important point. This chapter’s introduction gives formulas for $2 \times 2$ and $3 \times 3$ determinants, so we know that they exist, but not for determinant functions on matrices that are $4 \times 4$ or larger. Instead, Definition 2.1 gives properties that a determinant function should have and leads to computing determinants by Gauss’s Method.

However, for any matrix we can reduce it to echelon form by Gauss’s Method in multiple ways. For example, given a reduction we could change it by inserting a first step that multiplies the top row by $2$ and then a second step that multiplies it by $1 / 2$ . So we have to worry that two different Gauss’s Method reductions could lead to two different computed values for the determinant.

That is, we must verify that Definition 2.1 gives a well-defined function. The next two subsections do this, showing that there exists a well-defined function satisfying the definition.

But first we show that if there is such a function then there is no more than one. The example above illustrates the idea: we got $5$ by following the properties of the definition. So while we have not yet proved that $\underset{4 \times 4}{det}$ exists, that there is a function with properties (1) – (4), if such a function satisfying them does exist then we know what value it gives on the above matrix.

Lemma 2.7 For each $n$ , if there is an $n \times n$ determinant function then it is unique.

Proof Suppose that there are two functions $\underset{1}{det}, \underset{2}{det} : ℳ_{n \times n} \to ℝ$ satisfying the properties of Definition 2.1 and its consequence Lemma 2.4. Given a square matrix $M$ , fix some way of performing Gauss’s Method to bring the matrix to echelon form (it does not matter that there are multiple ways, just fix one of them). By using this fixed reduction as in the above examples— keeping track of row-scaling factors and how the sign alternates on row swaps, and then multiplying down the diagonal of the echelon form result— we can compute the value that these two functions must return on $M$ , and they must return the same value. Since they give the same output on every input, they are the same function.

QED

The ‘if there is an $n \times n$ determinant function’ emphasizes that, although we can use Gauss’s Method to compute the only value that a determinant function could possibly return, we haven’t yet shown that such a function exists for all $n$ . The rest of this section does that.

Exercises

For these, assume that an $n \times n$ determinant function exists for all $n$ .

Exercise 2.8 Supplied answer

Recommended. Find each determinant by performing one row operation.
1. $| \begin{matrix} 1 & - 2 & 1 & 2 \\ 2 & - 4 & 1 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 5 \end{matrix} |$
2. $| \begin{matrix} 1 & 1 & - 2 \\ 0 & 0 & 4 \\ 0 & 3 & - 6 \end{matrix} |$
Back to Exercise 2.8

Answer.
1. Do $2 ρ_{1} + ρ_{2}$ to get echelon form, and then multiply down the diagonal. The determinant is $0$ .
2. Swapping the second and third rows brings the system to echelon form (and changes the sign of the determinant). Multiplying down the diagonal gives $12$ , so the determinant of the given matrix is $- 12$ .
Exercise 2.9 Supplied answer

Recommended. Use Gauss’s Method to find each determinant.
1. $| \begin{matrix} 3 & 1 & 2 \\ 3 & 1 & 0 \\ 0 & 1 & 4 \end{matrix} |$
2. $| \begin{matrix} 1 & 0 & 0 & 1 \\ 2 & 1 & 1 & 0 \\ - 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 0 \end{matrix} |$
Back to Exercise 2.9

Answer.
1. $| \begin{matrix} 3 & 1 & 2 \\ 3 & 1 & 0 \\ 0 & 1 & 4 \end{matrix} | = | \begin{matrix} 3 & 1 & 2 \\ 0 & 0 & - 2 \\ 0 & 1 & 4 \end{matrix} | = - | \begin{matrix} 3 & 1 & 2 \\ 0 & 1 & 4 \\ 0 & 0 & - 2 \end{matrix} | = 6$
2. $| \begin{matrix} 1 & 0 & 0 & 1 \\ 2 & 1 & 1 & 0 \\ - 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 0 \end{matrix} | = | \begin{matrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & - 2 \\ 0 & 0 & 1 & 1 \\ 0 & 1 & 1 & - 1 \end{matrix} | = | \begin{matrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & - 2 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{matrix} | = 1$
Exercise 2.10 Supplied answer

Use Gauss’s Method to find each.
1. $| \begin{matrix} 2 & - 1 \\ - 1 & - 1 \end{matrix} |$
2. $| \begin{matrix} 1 & 1 & 0 \\ 3 & 0 & 2 \\ 5 & 2 & 2 \end{matrix} |$
Back to Exercise 2.10

Answer.
1. $| \begin{matrix} 2 & - 1 \\ - 1 & - 1 \end{matrix} | = | \begin{matrix} 2 & - 1 \\ 0 & - 3 / 2 \end{matrix} | = - 3$ ;
2. $| \begin{matrix} 1 & 1 & 0 \\ 3 & 0 & 2 \\ 5 & 2 & 2 \end{matrix} | = | \begin{matrix} 1 & 1 & 0 \\ 0 & - 3 & 2 \\ 0 & - 3 & 2 \end{matrix} | = | \begin{matrix} 1 & 1 & 0 \\ 0 & - 3 & 2 \\ 0 & 0 & 0 \end{matrix} | = 0$
Exercise 2.11 Supplied answer

For which values of $k$ does this system have a unique solution?

$\begin{array}{rcrcrcrcr} x & + & z & - & w & = & 2 \\ y & - & 2 z & = & 3 \\ x & + & k z & = & 4 \\ z & - & w & = & 2 \end{array}$

Back to Exercise 2.11

Answer. When is the determinant not zero?

$| \begin{matrix} 1 & 0 & 1 & - 1 \\ 0 & 1 & - 2 & 0 \\ 1 & 0 & k & 0 \\ 0 & 0 & 1 & - 1 \end{matrix} | = | \begin{matrix} 1 & 0 & 1 & - 1 \\ 0 & 1 & - 2 & 0 \\ 0 & 0 & k - 1 & 1 \\ 0 & 0 & 1 & - 1 \end{matrix} |$

Obviously, $k = 1$ gives nonsingularity and hence a nonzero determinant. If $k \neq 1$ then we get echelon form with a $(- 1 / k - 1) ρ_{3} + ρ_{4}$ combination.

$= | \begin{matrix} 1 & 0 & 1 & - 1 \\ 0 & 1 & - 2 & 0 \\ 0 & 0 & k - 1 & 1 \\ 0 & 0 & 0 & - 1 - (1 / k - 1) \end{matrix} |$

Multiplying down the diagonal gives $(k - 1) (- 1 - (1 / k - 1)) = - (k - 1) - 1 = - k$ . Thus the matrix has a nonzero determinant, and so the system has a unique solution, if and only if $k \neq 0$ .
Exercise 2.12 Supplied answer

Recommended. Express each of these in terms of $| H |$ .
1. $| \begin{matrix} h_{3, 1} & h_{3, 2} & h_{3, 3} \\ h_{2, 1} & h_{2, 2} & h_{2, 3} \\ h_{1, 1} & h_{1, 2} & h_{1, 3} \end{matrix} |$
2. $| \begin{matrix} - h_{1, 1} & - h_{1, 2} & - h_{1, 3} \\ - 2 h_{2, 1} & - 2 h_{2, 2} & - 2 h_{2, 3} \\ - 3 h_{3, 1} & - 3 h_{3, 2} & - 3 h_{3, 3} \end{matrix} |$
3. $| \begin{matrix} h_{1, 1} + h_{3, 1} & h_{1, 2} + h_{3, 2} & h_{1, 3} + h_{3, 3} \\ h_{2, 1} & h_{2, 2} & h_{2, 3} \\ 5 h_{3, 1} & 5 h_{3, 2} & 5 h_{3, 3} \end{matrix} |$
Back to Exercise 2.12

Answer.
1. Condition (2) of the definition of determinants applies via the swap $ρ_{1} \leftrightarrow ρ_{3}$ .
  
  $| \begin{matrix} h_{3, 1} & h_{3, 2} & h_{3, 3} \\ h_{2, 1} & h_{2, 2} & h_{2, 3} \\ h_{1, 1} & h_{1, 2} & h_{1, 3} \end{matrix} | = - | \begin{matrix} h_{1, 1} & h_{1, 2} & h_{1, 3} \\ h_{2, 1} & h_{2, 2} & h_{2, 3} \\ h_{3, 1} & h_{3, 2} & h_{3, 3} \end{matrix} |$
2. Condition (3) applies.
  
  $\begin{matrix} | \begin{matrix} - h_{1, 1} & - h_{1, 2} & - h_{1, 3} \\ - 2 h_{2, 1} & - 2 h_{2, 2} & - 2 h_{2, 3} \\ - 3 h_{3, 1} & - 3 h_{3, 2} & - 3 h_{3, 3} \end{matrix} | = (- 1) \cdot (- 2) \cdot (- 3) \cdot | \begin{matrix} h_{1, 1} & h_{1, 2} & h_{1, 3} \\ h_{2, 1} & h_{2, 2} & h_{2, 3} \\ h_{3, 1} & h_{3, 2} & h_{3, 3} \end{matrix} | \\ = (- 6) \cdot | \begin{matrix} h_{1, 1} & h_{1, 2} & h_{1, 3} \\ h_{2, 1} & h_{2, 2} & h_{2, 3} \\ h_{3, 1} & h_{3, 2} & h_{3, 3} \end{matrix} | \end{matrix}$
3. $\begin{matrix} | \begin{matrix} h_{1, 1} + h_{3, 1} & h_{1, 2} + h_{3, 2} & h_{1, 3} + h_{3, 3} \\ h_{2, 1} & h_{2, 2} & h_{2, 3} \\ 5 h_{3, 1} & 5 h_{3, 2} & 5 h_{3, 3} \end{matrix} | \\ \begin{aligned} = 5 \cdot | \begin{array}{c} h_{1, 1} + h_{3, 1} & h_{1, 2} + h_{3, 2} & h_{1, 3} + h_{3, 3} \\ h_{2, 1} & h_{2, 2} & h_{2, 3} \\ h_{3, 1} & h_{3, 2} & h_{3, 3} \end{array} | \\ = 5 \cdot | \begin{array}{c} h_{1, 1} & h_{1, 2} & h_{1, 3} \\ h_{2, 1} & h_{2, 2} & h_{2, 3} \\ h_{3, 1} & h_{3, 2} & h_{3, 3} \end{array} | \end{aligned} \end{matrix}$
Exercise 2.13 Supplied answer

Recommended. Find the determinant of a diagonal matrix.

Back to Exercise 2.13

Answer. A diagonal matrix is in echelon form, so the determinant is the product down the diagonal.
Exercise 2.14 Supplied answer

Describe the solution set of a homogeneous linear system if the determinant of the matrix of coefficients is nonzero.

Back to Exercise 2.14

Answer. It is the trivial subspace.
Exercise 2.15 Supplied answer

Recommended. Show that this determinant is zero.

$| \begin{matrix} y + z & x + z & x + y \\ x & y & z \\ 1 & 1 & 1 \end{matrix} |$

Back to Exercise 2.15

Answer. Adding the second row to the first gives a matrix whose first row is $x + y + z$ times its third row.
Exercise 2.16 Supplied answer
1. Find the $1 \times 1$ , $2 \times 2$ , and $3 \times 3$ matrices with $i, j$ entry given by $(- 1)^{i + j}$ .
2. Find the determinant of the square matrix with $i, j$ entry $(- 1)^{i + j}$ .
Back to Exercise 2.16

Answer.
1. $(\begin{matrix} 1 \end{matrix})$ , $(\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix})$ , $(\begin{matrix} 1 & - 1 & 1 \\ - 1 & 1 & - 1 \\ 1 & - 1 & 1 \end{matrix})$
2. The determinant in the $1 \times 1$ case is $1$ . In every other case the second row is the negative of the first, and so matrix is singular and the determinant is zero.
Exercise 2.17 Supplied answer
1. Find the $1 \times 1$ , $2 \times 2$ , and $3 \times 3$ matrices with $i, j$ entry given by $i + j$ .
2. Find the determinant of the square matrix with $i, j$ entry $i + j$ .
Back to Exercise 2.17

Answer.
1. $(\begin{matrix} 2 \end{matrix})$ , $(\begin{matrix} 2 & 3 \\ 3 & 4 \end{matrix})$ , $(\begin{matrix} 2 & 3 & 4 \\ 3 & 4 & 5 \\ 4 & 5 & 6 \end{matrix})$
2. The $1 \times 1$ and $2 \times 2$ cases yield these.
  
  $| \begin{matrix} 2 \end{matrix} | = 2 | \begin{matrix} 2 & 3 \\ 3 & 4 \end{matrix} | = - 1$
  
  And $n \times n$ matrices with $n \geq 3$ are singular, e.g.,
  
  $| \begin{matrix} 2 & 3 & 4 \\ 3 & 4 & 5 \\ 4 & 5 & 6 \end{matrix} | = 0$
  
  because twice the second row minus the first row equals the third row. Checking this is routine.
Exercise 2.18 Supplied answer

Recommended. Show that determinant functions are not linear by giving a case where $| A + B | \neq | A | + | B |$ .

Back to Exercise 2.18

Answer. This one

$A = B = (\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix})$

is easy to check.

$| A + B | = | \begin{matrix} 2 & 4 \\ 6 & 8 \end{matrix} | = - 8 | A | + | B | = - 2 - 2 = - 4$

By the way, this also gives an example where scalar multiplication is not preserved $| 2 \cdot A | \neq 2 \cdot | A |$ .
Exercise 2.19 Supplied answer

The second condition in the definition, that row swaps change the sign of a determinant, is somewhat annoying. It means we have to keep track of the number of swaps, to compute how the sign alternates. Can we get rid of it? Can we replace it with the condition that row swaps leave the determinant unchanged? (If so then we would need new $1 \times 1$ , $2 \times 2$ , and $3 \times 3$ formulas, but that would be a minor matter.)

Back to Exercise 2.19

Answer. No, we cannot replace it. Remark 2.2 shows that the four conditions after the replacement would conflict —no function satisfies all four.
Exercise 2.20 Supplied answer

Prove that the determinant of any triangular matrix, upper or lower, is the product down its diagonal.

Back to Exercise 2.20

Answer. A upper-triangular matrix is in echelon form.

A lower-triangular matrix is either singular or nonsingular. If it is singular then it has a zero on its diagonal and so its determinant (namely, zero) is indeed the product down its diagonal. If it is nonsingular then it has no zeroes on its diagonal, and we can reduce it by Gauss’s Method to echelon form without changing the diagonal.
Exercise 2.21 Supplied answer

Refer to the definition of elementary matrices in the Mechanics of Matrix Multiplication subsection.
1. What is the determinant of each kind of elementary matrix?
2. Prove that if $E$ is any elementary matrix then $| E S | = | E | | S |$ for any appropriately sized $S$ .
3. (This question doesn’t involve determinants.) Prove that if $T$ is singular then a product $T S$ is also singular.
4. Show that $| T S | = | T | | S |$ .
5. Show that if $T$ is nonsingular then $| T^{- 1} | = {| T |}^{- 1}$ .
Back to Exercise 2.21

Answer.
1. The properties in the definition of determinant show that $| M_{i} (k) | = k$ , $| P_{i, j} | = - 1$ , and $| C_{i, j} (k) | = 1$ .
2. The three cases are easy to check by recalling the action of left multiplication by each type of matrix.
3. If $T S$ is invertible $(T S) M = I$ then the associative property of matrix multiplication $T (S M) = I$ shows that $T$ is invertible. So if $T$ is not invertible then neither is $T S$ .
4. If $T$ is singular then apply the prior answer: $| T S | = 0$ and $| T | \cdot | S | = 0 \cdot | S | = 0$ . If $T$ is not singular then we can write it as a product of elementary matrices $| T S | = | E_{r} \dots E_{1} S | = | E_{r} | \dots | E_{1} | \cdot | S | = | E_{r} \dots E_{1} | | S | = | T | | S |$ .
5. $1 = | I | = | T \cdot T^{- 1} | = | T | | T^{- 1} |$
Exercise 2.22 Supplied answer

Prove that the determinant of a product is the product of the determinants $| T S | = | T | | S |$ in this way. Fix the $n \times n$ matrix $S$ and consider the function $d : ℳ_{n \times n} \to ℝ$ given by $T \mapsto | T S | / | S |$ .
1. Check that $d$ satisfies condition (1) in the definition of a determinant function.
2. Check condition (2).
3. Check condition (3).
4. Check condition (4).
5. Conclude the determinant of a product is the product of the determinants.
Back to Exercise 2.22

Answer.
1. We must show that if
  
  $T ⟶_{\begin{array}{l} \end{array}}^{k ρ_{i} + ρ_{j}} \hat{T}$
  
  then $d (T) = | T S | / | S | = | \hat{T} S | / | S | = d (\hat{T})$ . We will be done if we show that combining rows first and then multiplying to get $\hat{T} S$ gives the same result as multiplying first to get $T S$ and then combining (because the determinant $| T S |$ is unaffected by the combination so we’ll then have $| \hat{T} S | = | T S |$ , and hence $d (\hat{T}) = d (T)$ ). That argument runs: after adding $k$ times row $i$ of $T S$ to row $j$ of $T S$ , the $j, p$ entry is $(k t_{i, 1} + t_{j, 1}) s_{1, p} + \dots + (k t_{i, r} + t_{j, r}) s_{r, p}$ , which is the $j, p$ entry of $\hat{T} S$ .
2. We need only show that swapping $T ⟶_{\begin{array}{l} \end{array}}^{ρ_{i} \leftrightarrow ρ_{j}} \hat{T}$ and then multiplying to get $\hat{T} S$ gives the same result as multiplying $T$ by $S$ and then swapping (because, as the determinant $| T S |$ changes sign on the row swap, we’ll then have $| \hat{T} S | = - | T S |$ , and so $d (\hat{T}) = - d (T)$ ). That argument runs just like the prior one.
3. Not surprisingly by now, we need only show that multiplying a row by a scalar $T ⟶_{\begin{array}{l} \end{array}}^{k ρ_{i}} \hat{T}$ and then computing $\hat{T} S$ gives the same result as first computing $T S$ and then multiplying the row by $k$ (as the determinant $| T S |$ is rescaled by $k$ the multiplication, we’ll have $| \hat{T} S | = k | T S |$ , so $d (\hat{T}) = k d (T)$ ). The argument runs just as above.
4. Clear.
5. Because we’ve shown that $d (T)$ is a determinant and that determinant functions (if they exist) are unique, we have that so $| T | = d (T) = | T S | / | S |$ .
Exercise 2.23 Supplied answer

A submatrix of a given matrix $A$ is one that we get by deleting some of the rows and columns of $A$ . Thus, the first matrix here is a submatrix of the second.

$(\begin{matrix} 3 & 1 \\ 2 & 5 \end{matrix}) (\begin{matrix} 3 & 4 & 1 \\ 0 & 9 & - 2 \\ 2 & - 1 & 5 \end{matrix})$

Prove that for any square matrix, the rank of the matrix is $r$ if and only if $r$ is the largest integer such that there is an $r \times r$ submatrix with a nonzero determinant.

Back to Exercise 2.23

Answer. We will first argue that a rank $r$ matrix has a $r \times r$ submatrix with nonzero determinant. A rank $r$ matrix has a linearly independent set of $r$ rows. A matrix made from those rows will have row rank $r$ and thus has column rank $r$ . Conclusion: from those $r$ rows we can extract a linearly independent set of $r$ columns, and so the original matrix has a $r \times r$ submatrix of rank $r$ .

We finish by showing that if $r$ is the largest such integer then the rank of the matrix is $r$ . We need only show, by the maximality of $r$ , that if a matrix has a $k \times k$ submatrix of nonzero determinant then the rank of the matrix is at least $k$ . Consider such a $k \times k$ submatrix. Its rows are parts of the rows of the original matrix, clearly the set of whole rows is linearly independent. Thus the row rank of the original matrix is at least $k$ , and the row rank of a matrix equals its rank.
Exercise 2.24 Supplied answer

Prove that a matrix with rational entries has a rational determinant.

Back to Exercise 2.24

Answer. A matrix with only rational entries reduces with Gauss’s Method to an echelon form matrix using only rational arithmetic. Thus the entries on the diagonal must be rationals, and so the product down the diagonal is rational.
Exercise 2.25 Supplied answer

Puzzle. [Am. Math. Mon., Feb. 1953] Find the element of likeness in (a) simplifying a fraction, (b) powdering the nose, (c) building new steps on the church, (d) keeping emeritus professors on campus, (e) putting $B$ , $C$ , $D$ in the determinant

$| \begin{matrix} 1 & a & a^{2} & a^{3} \\ a^{3} & 1 & a & a^{2} \\ B & a^{3} & 1 & a \\ C & D & a^{3} & 1 \end{matrix} | .$

Back to Exercise 2.25

Answer. This is how the answer was given in the cited source. The value $(1 - a^{4})^{3}$ of the determinant is independent of the values $B$ , $C$ , $D$ . Hence operation (e) does not change the value of the determinant but merely changes its appearance. Thus the element of likeness in (a), (b), (c), (d), and (e) is only that the appearance of the principle entity is changed. The same element appears in (f) changing the name-label of a rose, (g) writing a decimal integer in the scale of $12$ , (h) gilding the lily, (i) whitewashing a politician, and (j) granting an honorary degree.

The Permutation Expansion

The prior subsection defines a function to be a determinant if it satisfies four conditions and shows that there is at most one $n \times n$ determinant function for each $n$ . What is left is to show that for each $n$ such a function exists.

But, we easily compute determinants: we use Gauss’s Method, keeping track of the sign changes from row swaps, and end by multiplying down the diagonal. How could they not exist?

The difficulty is to show that the computation gives a well-defined—that is, unique—result. Consider these two Gauss’s Method reductions of the same matrix, the first without any row swap

$(\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}) ⟶_{\begin{array}{l} \end{array}}^{- 3 ρ_{1} + ρ_{2}} (\begin{matrix} 1 & 2 \\ 0 & - 2 \end{matrix})$

and the second with one.

$(\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}) ⟶_{\begin{array}{l} \end{array}}^{ρ_{1} \leftrightarrow ρ_{2}} (\begin{matrix} 3 & 4 \\ 1 & 2 \end{matrix}) ⟶_{\begin{array}{l} \end{array}}^{- (1 / 3) ρ_{1} + ρ_{2}} (\begin{matrix} 3 & 4 \\ 0 & 2 / 3 \end{matrix})$

Both yield the determinant $- 2$ since in the second one we note that the row swap changes the sign of the result we get by multiplying down the diagonal. The fact that we are able to proceed in two ways opens the possibility that the two give different answers. That is, the way that we have given to compute determinant values does not plainly eliminate the possibility that there might be, say, two reductions of some $7 \times 7$ matrix that lead to different determinant values. In that case we would not have a function, since the definition of a function is that for each input there must be exactly associated one output. The rest of this section shows that the definition Definition 2.1 never leads to a conflict.

To do this we will define an alternative way to find the value of a determinant. (This alternative is less useful in practice because it is slow. But it is very useful for theory.) The key idea is that condition (3) of Definition 2.1 shows that the determinant function is not linear.

Example 3.1 With condition (3) scalars come out of each row separately,

$| \begin{matrix} 4 & 2 \\ - 2 & 6 \end{matrix} | = 2 \cdot | \begin{matrix} 2 & 1 \\ - 2 & 6 \end{matrix} | = 4 \cdot | \begin{matrix} 2 & 1 \\ - 1 & 3 \end{matrix} |$

not from the entire matrix at once. So, where

$A = (\begin{matrix} 2 & 1 \\ - 1 & 3 \end{matrix})$

then $det (2 A) \neq 2 \cdot det (A)$ (instead, $det (2 A) = 4 \cdot det (A)$ ).

Since scalars come out a row at a time we might guess that determinants are linear a row at a time.

Definition 3.2 Let $V$ be a vector space. A map $f : V^{n} \to ℝ$ is multilinear if

$f ({\vec{ρ}}_{1}, \dots, \vec{v} + \vec{w}, \dots, {\vec{ρ}}_{n}) = f ({\vec{ρ}}_{1}, \dots, \vec{v}, \dots, {\vec{ρ}}_{n}) + f ({\vec{ρ}}_{1}, \dots, \vec{w}, \dots, {\vec{ρ}}_{n})$
$f ({\vec{ρ}}_{1}, \dots, k \vec{v}, \dots, {\vec{ρ}}_{n}) = k \cdot f ({\vec{ρ}}_{1}, \dots, \vec{v}, \dots, {\vec{ρ}}_{n})$

for $\vec{v}, \vec{w} \in V$ and $k \in ℝ$ .

Lemma 3.3 Determinants are multilinear.

Proof Property (2) here is just Definition 2.1’s condition (3) so we need only verify property (1).

There are two cases. If the set of other rows ${{\vec{ρ}}_{1}, \dots, {\vec{ρ}}_{i - 1}, {\vec{ρ}}_{i + 1}, \dots, {\vec{ρ}}_{n}}$ is linearly dependent then all three matrices are singular and so all three determinants are zero and the equality is trivial.

Therefore assume that the set of other rows is linearly independent. We can make a basis by adding one more vector $⟨ {\vec{ρ}}_{1}, \dots, {\vec{ρ}}_{i - 1}, \vec{β}, {\vec{ρ}}_{i + 1}, \dots, {\vec{ρ}}_{n} ⟩$ . Express $\vec{v}$ and $\vec{w}$ with respect to this basis

$\begin{aligned} \vec{v} & = v_{1} {\vec{ρ}}_{1} + \dots + v_{i - 1} {\vec{ρ}}_{i - 1} + v_{i} \vec{β} + v_{i + 1} {\vec{ρ}}_{i + 1} + \dots + v_{n} {\vec{ρ}}_{n} \\ \vec{w} & = w_{1} {\vec{ρ}}_{1} + \dots + w_{i - 1} {\vec{ρ}}_{i - 1} + w_{i} \vec{β} + w_{i + 1} {\vec{ρ}}_{i + 1} + \dots + w_{n} {\vec{ρ}}_{n} \end{aligned}$

and add.

$\vec{v} + \vec{w} = (v_{1} + w_{1}) {\vec{ρ}}_{1} + \dots + (v_{i} + w_{i}) \vec{β} + \dots + (v_{n} + w_{n}) {\vec{ρ}}_{n}$

Consider the left side of (1) and expand $\vec{v} + \vec{w}$ .

$\begin{matrix} det ({\vec{ρ}}_{1}, \dots, (v_{1} + w_{1}) {\vec{ρ}}_{1} + \dots + (v_{i} + w_{i}) \vec{β} + \dots + (v_{n} + w_{n}) {\vec{ρ}}_{n}, \dots, {\vec{ρ}}_{n}) & (*) \end{matrix}$

By the definition of determinant’s condition (1), the value of ( $*$ ) is unchanged by the operation of adding $- (v_{1} + w_{1}) {\vec{ρ}}_{1}$ to the $i$ -th row $\vec{v} + \vec{w}$ . The $i$ -th row becomes this.

$\vec{v} + \vec{w} - (v_{1} + w_{1}) {\vec{ρ}}_{1} = (v_{2} + w_{2}) {\vec{ρ}}_{2} + \dots + (v_{i} + w_{i}) \vec{β} + \dots + (v_{n} + w_{n}) {\vec{ρ}}_{n}$

Next add $- (v_{2} + w_{2}) {\vec{ρ}}_{2}$ , etc., to eliminate all of the terms from the other rows. Apply condition (3) from the definition of determinant.

$\begin{matrix} det ({\vec{ρ}}_{1}, \dots, \vec{v} + \vec{w}, \dots, {\vec{ρ}}_{n}) \\ \begin{aligned} = det ({\vec{ρ}}_{1}, \dots, (v_{i} + w_{i}) \cdot \vec{β}, \dots, {\vec{ρ}}_{n}) \\ = (v_{i} + w_{i}) \cdot det ({\vec{ρ}}_{1}, \dots, \vec{β}, \dots, {\vec{ρ}}_{n}) \\ = v_{i} \cdot det ({\vec{ρ}}_{1}, \dots, \vec{β}, \dots, {\vec{ρ}}_{n}) + w_{i} \cdot det ({\vec{ρ}}_{1}, \dots, \vec{β}, \dots, {\vec{ρ}}_{n}) \end{aligned} \end{matrix}$

Now this is a sum of two determinants. To finish, bring $v_{i}$ and $w_{i}$ back inside in front of the $\vec{β}$ ’s and use row combinations again, this time to reconstruct the expressions of $\vec{v}$ and $\vec{w}$ in terms of the basis. That is, start with the operations of adding $v_{1} {\vec{ρ}}_{1}$ to $v_{i} \vec{β}$ and $w_{1} {\vec{ρ}}_{1}$ to $w_{i} {\vec{ρ}}_{1}$ , etc., to get the expansions of $\vec{v}$ and $\vec{w}$ .

QED

Multilinearity allows us to expand a determinant into a sum of determinants, each of which involves a simple matrix.

Example 3.4 Use property (1) of multilinearity to break up the first row

$| \begin{matrix} 2 & 1 \\ 4 & 3 \end{matrix} | = | \begin{matrix} 2 & 0 \\ 4 & 3 \end{matrix} | + | \begin{matrix} 0 & 1 \\ 4 & 3 \end{matrix} |$

and then use (1) again to break each along the second row.

$= | \begin{matrix} 2 & 0 \\ 4 & 0 \end{matrix} | + | \begin{matrix} 2 & 0 \\ 0 & 3 \end{matrix} | + | \begin{matrix} 0 & 1 \\ 4 & 0 \end{matrix} | + | \begin{matrix} 0 & 1 \\ 0 & 3 \end{matrix} |$

The result is four determinants. In each row of each of the four there is a single entry from the original matrix.

Example 3.5 In the same way, a $3 \times 3$ determinant separates into a sum of many simpler determinants. Splitting along the first row produces three determinants (we have highlighted the zero in the $1, 3$ position to set it off visually from the zeroes that appear as part of the splitting).

$| \begin{matrix} 2 & 1 & - 1 \\ 4 & 3 & 0 \\ 2 & 1 & 5 \end{matrix} | = | \begin{matrix} 2 & 0 & 0 \\ 4 & 3 & 0 \\ 2 & 1 & 5 \end{matrix} | + | \begin{matrix} 0 & 1 & 0 \\ 4 & 3 & 0 \\ 2 & 1 & 5 \end{matrix} | + | \begin{matrix} 0 & 0 & - 1 \\ 4 & 3 & 0 \\ 2 & 1 & 5 \end{matrix} |$

In turn, each of the above splits in three along the second row. Then each of the nine splits in three along the third row. The result is twenty seven determinants, such that each row contains a single entry from the starting matrix.

$= | \begin{matrix} 2 & 0 & 0 \\ 4 & 0 & 0 \\ 2 & 0 & 0 \end{matrix} | + | \begin{matrix} 2 & 0 & 0 \\ 4 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} | + | \begin{matrix} 2 & 0 & 0 \\ 4 & 0 & 0 \\ 0 & 0 & 5 \end{matrix} | + | \begin{matrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 2 & 0 & 0 \end{matrix} | + \dots + | \begin{matrix} 0 & 0 & - 1 \\ 0 & 0 & 0 \\ 0 & 0 & 5 \end{matrix} |$

So multilinearity will expand an $n \times n$ determinant into a sum of $n^{n}$ -many determinants, where each row of each determinant contains a single entry from the starting matrix.

In this expansion, although there are lots of terms, most of them have a determinant of zero.

Example 3.6 In each of these examples from the prior expansion, two of the entries from the original matrix are in the same column.

$| \begin{matrix} 2 & 0 & 0 \\ 4 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} | | \begin{matrix} 0 & 0 & - 1 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{matrix} | | \begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 5 \end{matrix} |$

For instance, in the first matrix the $2$ and the $4$ both come from the first column of the original matrix. In the second matrix the $- 1$ and $5$ both come from the third column. And in the third matrix the $0$ and $5$ both come from the third column. Any such matrix is singular because one row is a multiple of the other. Thus any such determinant is zero, by Lemma 2.4.

With that observation the above expansion of the $3 \times 3$ determinant into the sum of the twenty seven determinants simplifies to the sum of these six where the entries from the original matrix come one per row, and also one per column.

$\begin{aligned} | \begin{array}{c} 2 & 1 & - 1 \\ 4 & 3 & 0 \\ 2 & 1 & 5 \end{array} | & = | \begin{array}{c} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \end{array} | + | \begin{array}{c} 2 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \end{array} | \\ + | \begin{array}{c} 0 & 1 & 0 \\ 4 & 0 & 0 \\ 0 & 0 & 5 \end{array} | + | \begin{array}{c} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 2 & 0 & 0 \end{array} | \\ + | \begin{array}{c} 0 & 0 & - 1 \\ 4 & 0 & 0 \\ 0 & 1 & 0 \end{array} | + | \begin{array}{c} 0 & 0 & - 1 \\ 0 & 3 & 0 \\ 2 & 0 & 0 \end{array} | \end{aligned}$

In that expansion we can bring out the scalars.

$\begin{aligned} = (2) (3) (5) | \begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} | + (2) (0) (1) | \begin{array}{c} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array} | \\ + (1) (4) (5) | \begin{array}{c} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array} | + (1) (0) (2) | \begin{array}{c} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array} | \\ + (- 1) (4) (1) | \begin{array}{c} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array} | + (- 1) (3) (2) | \begin{array}{c} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} | \end{aligned}$

To finish, evaluate those six determinants by row-swapping them to the identity matrix, keeping track of the sign changes.

$\begin{aligned} = 30 \cdot (+ 1) + 0 \cdot (- 1) \\ + 20 \cdot (- 1) + 0 \cdot (+ 1) \\ - 4 \cdot (+ 1) - 6 \cdot (- 1) = 12 \end{aligned}$

That example captures this subsection’s new calculation scheme. Multilinearity expands a determinant into many separate determinants, each with one entry from the original matrix per row. Most of these have one row that is a multiple of another so we omit them. We are left with the determinants that have one entry per row and column from the original matrix. Factoring out the scalars further reduces the determinants that we must compute to the one-entry-per-row-and-column matrices where all entries are $1$ ’s.

Recall Definition Three.IV.3.14, that a permutation matrix is square, with entries $0$ ’s except for a single $1$ in each row and column. We now introduce a notation for permutation matrices.

Definition 3.7 An $n$ -permutation is a function on the first $n$ positive integers $ϕ : {1, \dots, n} \to {1, \dots, n}$ that is one-to-one and onto.

In a permutation each number $1$ , …, $n$ appears as output for one and only one input. We can denote a permutation as a sequence $ϕ = ⟨ ϕ (1), ϕ (2), \dots, ϕ (n) ⟩$ .

Example 3.8 The $2$ -permutations are the functions $ϕ_{1} : {1, 2} \to {1, 2}$ given by $ϕ_{1} (1) = 1$ , $ϕ_{1} (2) = 2$ , and $ϕ_{2} : {1, 2} \to {1, 2}$ given by $ϕ_{2} (1) = 2$ , $ϕ_{2} (2) = 1$ . The sequence notation is shorter: $ϕ_{1} = ⟨ 1, 2 ⟩$ and $ϕ_{2} = ⟨ 2, 1 ⟩$ .

Example 3.9 In the sequence notation the $3$ -permutations are $ϕ_{1} = ⟨ 1, 2, 3 ⟩$ , $ϕ_{2} = ⟨ 1, 3, 2 ⟩$ , $ϕ_{3} = ⟨ 2, 1, 3 ⟩$ , $ϕ_{4} = ⟨ 2, 3, 1 ⟩$ , $ϕ_{5} = ⟨ 3, 1, 2 ⟩$ , and $ϕ_{6} = ⟨ 3, 2, 1 ⟩$ .

We denote the row vector that is all $0$ ’s except for a $1$ in entry $j$ with $ι_{j}$ so that the four-wide $ι_{2}$ is $(\begin{matrix} 0 & 1 & 0 & 0 \end{matrix})$ . Now our notation for permutation matrices is: with any $ϕ = ⟨ ϕ (1), \dots, ϕ (n) ⟩$ associate the matrix whose rows are $ι_{ϕ (1)}$ , …, $ι_{ϕ (n)}$ . For instance, associated with the $4$ -permutation $ϕ = ⟨ 3, 2, 1, 4 ⟩$ is the matrix whose rows are the corresponding $ι$ ’s.

$P_{ϕ} = (\begin{matrix} ι_{3} \\ ι_{2} \\ ι_{1} \\ ι_{4} \end{matrix}) = (\begin{matrix} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix})$

Example 3.10 These are the permutation matrices for the $2$ -permutations listed in Example 3.8.

$P_{ϕ_{1}} = (\begin{matrix} ι_{1} \\ ι_{2} \end{matrix}) = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}) P_{ϕ_{2}} = (\begin{matrix} ι_{2} \\ ι_{1} \end{matrix}) = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix})$

For instance, $P_{ϕ_{2}}$ ’s first row is $ι_{ϕ_{2} (1)} = ι_{2}$ and its second is $ι_{ϕ_{2} (2)} = ι_{1}$ .

Example 3.11 Consider the $3$ -permutation $ϕ_{5} = ⟨ 3, 1, 2 ⟩$ . The permutation matrix $P_{ϕ_{5}}$ has rows $ι_{ϕ_{5} (1)} = ι_{3}$ , $ι_{ϕ_{5} (2)} = ι_{1}$ , and $ι_{ϕ_{5} (3)} = ι_{2}$ .

$P_{ϕ_{5}} = (\begin{matrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix})$

Definition 3.12 The permutation expansion for determinants is

$| \begin{matrix} t_{1, 1} & t_{1, 2} & \dots & t_{1, n} \\ t_{2, 1} & t_{2, 2} & \dots & t_{2, n} \\ ⋮ \\ t_{n, 1} & t_{n, 2} & \dots & t_{n, n} \end{matrix} | = \begin{array}{l} t_{1, ϕ_{1} (1)} t_{2, ϕ_{1} (2)} \dots t_{n, ϕ_{1} (n)} | P_{ϕ_{1}} | \\ + t_{1, ϕ_{2} (1)} t_{2, ϕ_{2} (2)} \dots t_{n, ϕ_{2} (n)} | P_{ϕ_{2}} | \\ ⋮ \\ + t_{1, ϕ_{k} (1)} t_{2, ϕ_{k} (2)} \dots t_{n, ϕ_{k} (n)} | P_{ϕ_{k}} | \end{array}$

where $ϕ_{1}, \dots, ϕ_{k}$ are all of the $n$ -permutations.

We can restate the formula in summation notation

$| T | = \sum_{permutations ϕ} t_{1, ϕ (1)} t_{2, ϕ (2)} \dots t_{n, ϕ (n)} | P_{ϕ} |$

read aloud as, “the sum, over all permutations $ϕ$ , of terms having the form $t_{1, ϕ (1)} t_{2, ϕ (2)} \dots t_{n, ϕ (n)} | P_{ϕ} |$ .”

Example 3.13 The familiar $2 \times 2$ determinant formula follows from the above

$\begin{aligned} | \begin{array}{c} t_{1, 1} & t_{1, 2} \\ t_{2, 1} & t_{2, 2} \end{array} | & = t_{1, 1} t_{2, 2} \cdot | P_{ϕ_{1}} | + t_{1, 2} t_{2, 1} \cdot | P_{ϕ_{2}} | \\ = t_{1, 1} t_{2, 2} \cdot | \begin{array}{c} 1 & 0 \\ 0 & 1 \end{array} | + t_{1, 2} t_{2, 1} \cdot | \begin{array}{c} 0 & 1 \\ 1 & 0 \end{array} | \\ = t_{1, 1} t_{2, 2} - t_{1, 2} t_{2, 1} \end{aligned}$

as does the $3 \times 3$ formula.

$\begin{aligned} | \begin{array}{c} t_{1, 1} & t_{1, 2} & t_{1, 3} \\ t_{2, 1} & t_{2, 2} & t_{2, 3} \\ t_{3, 1} & t_{3, 2} & t_{3, 3} \end{array} | & = \begin{aligned} t_{1, 1} t_{2, 2} t_{3, 3} | P_{ϕ_{1}} | + t_{1, 1} t_{2, 3} t_{3, 2} | P_{ϕ_{2}} | + t_{1, 2} t_{2, 1} t_{3, 3} | P_{ϕ_{3}} | \\ + t_{1, 2} t_{2, 3} t_{3, 1} | P_{ϕ_{4}} | + t_{1, 3} t_{2, 1} t_{3, 2} | P_{ϕ_{5}} | + t_{1, 3} t_{2, 2} t_{3, 1} | P_{ϕ_{6}} | \end{aligned} \\ = \begin{aligned} t_{1, 1} t_{2, 2} t_{3, 3} - t_{1, 1} t_{2, 3} t_{3, 2} - t_{1, 2} t_{2, 1} t_{3, 3} \\ + t_{1, 2} t_{2, 3} t_{3, 1} + t_{1, 3} t_{2, 1} t_{3, 2} - t_{1, 3} t_{2, 2} t_{3, 1} \end{aligned} \end{aligned}$

Computing a determinant with the permutation expansion typically takes longer than with Gauss’s Method. However, we will use it to prove that the determinant function exists. The proof is long so we will just state the result here and defer the proof to the following subsection.

Theorem 3.14 For each $n$ there is an $n \times n$ determinant function.

Also in the next subsection is the proof of the next result (they are together because the two proofs overlap).

Theorem 3.15 The determinant of a matrix equals the determinant of its transpose.

Because of this theorem, while we have so far stated determinant results in terms of rows, all of the results also hold in terms of columns.

Corollary 3.16 A matrix with two equal columns is singular. Column swaps change the sign of a determinant. Determinants are multilinear in their columns.

Proof For the first statement, transposing the matrix results in a matrix with the same determinant, and with two equal rows, and hence a determinant of zero. Prove the other two in the same way.

QED

We finish this subsection with a summary: determinant functions exist, are unique, and we know how to compute them. As for what determinants are about, perhaps these lines [Kemp] help make it memorable.

Determinant none,
Solution: lots or none.
Determinant some,
Solution: just one.

Exercises

This summarizes our notation for the $2$ - and $3$ -permutations.

$i$	$1$	$2$
$ϕ_{1} (i)$	$1$	$2$
$ϕ_{2} (i)$	$2$	$1$

$i$	$1$	$2$	$3$
$ϕ_{1} (i)$	$1$	$2$	$3$
$ϕ_{2} (i)$	$1$	$3$	$2$
$ϕ_{3} (i)$	$2$	$1$	$3$
$ϕ_{4} (i)$	$2$	$3$	$1$
$ϕ_{5} (i)$	$3$	$1$	$2$
$ϕ_{6} (i)$	$3$	$2$	$1$

Exercise 3.17 Supplied answer

Recommended. For this matrix, find the term associated with each $3$ -permutation.

$M = (\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{matrix})$

That is, fill in the rest of this table.

permutation $ϕ_{i}$	$ϕ_{1}$	$ϕ_{2}$	$ϕ_{3}$	$ϕ_{4}$	$ϕ_{5}$	$ϕ_{6}$
term $m_{1, ϕ_{i} (1)} m_{2, ϕ_{i} (2)} m_{3, ϕ_{i} (3)}$	$1 \cdot 5 \cdot 9$

Back to Exercise 3.17

Answer. Call the matrix $M$ .

permutation	$ϕ_{1}$	$ϕ_{2}$	$ϕ_{3}$	$ϕ_{4}$	$ϕ_{5}$	$ϕ_{6}$
term	$1 \cdot 5 \cdot 9$	$1 \cdot 6 \cdot 8$	$2 \cdot 4 \cdot 9$	$2 \cdot 6 \cdot 7$	$3 \cdot 4 \cdot 8$	$3 \cdot 5 \cdot 7$

Exercise 3.18 Supplied answer

Recommended. For each $3$ -permutation $ϕ$ find $| P_{ϕ} |$ .

Back to Exercise 3.18

Answer. We can swap each $P_{ϕ}$ to the identity matrix. If the number of swaps is even then it’s determinant is $+ 1$ , while if the number of swaps is odd then the determinant is $- 1$

permutation	$ϕ_{1}$	$ϕ_{2}$	$ϕ_{3}$	$ϕ_{4}$	$ϕ_{5}$	$ϕ_{6}$
number of swaps	$0$	$1$	$1$	$2$	$2$	$1$
$\| P_{ϕ_{i}} \|$	$+ 1$	$- 1$	$- 1$	$+ 1$	$+ 1$	$- 1$

Remark. In that table we simply swapped until we found a number that brought us back to ‘ $1, 2, 3$ ’. But is a different number of swaps possible? If one person found $2$ swaps and another found $4$ that would be OK, since both give a determinant of $+ 1$ . But if we can swap in two different ways and one of them is even and one is odd then that would be a problem. The next section shows that a mix of even and odd is not possible.

Exercise 3.19 Supplied answer

This determinant is $7$ by the $2 \times 2$ formula. Compute it with the permutation expansion.

$| \begin{matrix} 2 & 3 \\ 1 & 5 \end{matrix} |$

Back to Exercise 3.19

Answer.

$\begin{aligned} | \begin{array}{c} 2 & 3 \\ 1 & 5 \end{array} | & = 2 \cdot 5 \cdot | P_{ϕ_{1}} | + 1 \cdot 3 \cdot | P_{ϕ_{2}} | \\ = 2 \cdot 5 \cdot | \begin{array}{c} 1 & 0 \\ 0 & 1 \end{array} | + 1 \cdot 3 \cdot | \begin{array}{c} 0 & 1 \\ 1 & 0 \end{array} | \\ = 2 \cdot 5 \cdot 1 + 1 \cdot 3 \cdot (- 1) = 7 \end{aligned}$
Exercise 3.20 Supplied answer

This determinant is $0$ because the first two rows add to the third. Compute the determinant using the permutation expansion.

$| \begin{matrix} - 1 & 0 & 1 \\ 3 & 1 & 4 \\ 2 & 1 & 5 \end{matrix} |$

Back to Exercise 3.20

Answer.

$\begin{aligned} | \begin{array}{c} - 1 & 0 & 1 \\ 3 & 1 & 4 \\ 2 & 1 & 5 \end{array} | & = \begin{aligned} (- 1) (1) (5) | P_{ϕ_{1}} | + (- 1) (4) (1) | P_{ϕ_{2}} | + (0) (3) (5) | P_{ϕ_{3}} | \\ + (0) (4) (2) | P_{ϕ_{4}} | + (1) (3) (1) | P_{ϕ_{5}} | + (1) (1) (2) | P_{ϕ_{6}} | \end{aligned} \\ = \begin{aligned} (- 1) (1) (5) \cdot 1 + (- 1) (4) (1) \cdot (- 1) + (0) (3) (5) \cdot (- 1) \\ + (0) (4) (2) \cdot 1 + (1) (3) (1) \cdot 1 + (1) (1) (2) \cdot (- 1) \end{aligned} \\ = - 5 + 4 + 0 + 0 + 3 - 2 = 0 \end{aligned}$
Exercise 3.21 Supplied answer

Recommended. Compute the determinant by using the permutation expansion.
1. $| \begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{matrix} |$
2. $| \begin{matrix} 2 & 2 & 1 \\ 3 & - 1 & 0 \\ - 2 & 0 & 5 \end{matrix} |$
Back to Exercise 3.21

Answer.
1. This matrix is singular.
  
  $\begin{aligned} | \begin{array}{c} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array} | & = \begin{aligned} (1) (5) (9) | P_{ϕ_{1}} | + (1) (6) (8) | P_{ϕ_{2}} | + (2) (4) (9) | P_{ϕ_{3}} | \\ + (2) (6) (7) | P_{ϕ_{4}} | + (3) (4) (8) | P_{ϕ_{5}} | + (7) (5) (3) | P_{ϕ_{6}} | \end{aligned} \\ = 0 \end{aligned}$
2. This matrix is nonsingular.
  
  $\begin{aligned} | \begin{array}{c} 2 & 2 & 1 \\ 3 & - 1 & 0 \\ - 2 & 0 & 5 \end{array} | & = \begin{aligned} (2) (- 1) (5) | P_{ϕ_{1}} | + (2) (0) (0) | P_{ϕ_{2}} | + (2) (3) (5) | P_{ϕ_{3}} | \\ + (2) (0) (- 2) | P_{ϕ_{4}} | + (1) (3) (0) | P_{ϕ_{5}} | + (- 2) (- 1) (1) | P_{ϕ_{6}} | \end{aligned} \\ = - 42 \end{aligned}$
Exercise 3.22 Supplied answer

Recommended. Compute these both with Gauss’s Method and the permutation expansion formula.
1. $| \begin{matrix} 2 & 1 \\ 3 & 1 \end{matrix} |$
2. $| \begin{matrix} 0 & 1 & 4 \\ 0 & 2 & 3 \\ 1 & 5 & 1 \end{matrix} |$
Back to Exercise 3.22

Answer.
1. Gauss’s Method gives this
  
  $| \begin{matrix} 2 & 1 \\ 3 & 1 \end{matrix} | = | \begin{matrix} 2 & 1 \\ 0 & - 1 / 2 \end{matrix} | = - 1$
  
  and permutation expansion gives this.
  
  $| \begin{matrix} 2 & 1 \\ 3 & 1 \end{matrix} | = | \begin{matrix} 2 & 0 \\ 0 & 1 \end{matrix} | + | \begin{matrix} 0 & 1 \\ 3 & 0 \end{matrix} | = (2) (1) | \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} | + (1) (3) | \begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix} | = - 1$
2. Gauss’s Method gives this
  
  $| \begin{matrix} 0 & 1 & 4 \\ 0 & 2 & 3 \\ 1 & 5 & 1 \end{matrix} | = - | \begin{matrix} 1 & 5 & 1 \\ 0 & 2 & 3 \\ 0 & 1 & 4 \end{matrix} | = - | \begin{matrix} 1 & 5 & 1 \\ 0 & 2 & 3 \\ 0 & 0 & 5 / 2 \end{matrix} | = - 5$
  
  and the permutation expansion gives this.
  
  $\begin{aligned} | \begin{array}{c} 0 & 1 & 4 \\ 0 & 2 & 3 \\ 1 & 5 & 1 \end{array} | & = \begin{aligned} (0) (2) (1) | P_{ϕ_{1}} | + (0) (3) (5) | P_{ϕ_{2}} | + (1) (0) (1) | P_{ϕ_{3}} | \\ + (1) (3) (1) | P_{ϕ_{4}} | + (4) (0) (5) | P_{ϕ_{5}} | + (4) (2) (1) | P_{ϕ_{6}} | \end{aligned} \\ = - 5 \end{aligned}$
Exercise 3.23 Supplied answer

Recommended. Use the permutation expansion formula to derive the formula for $3 \times 3$ determinants.

Back to Exercise 3.23

Answer. Following Example 3.6 gives this.

$\begin{aligned} | \begin{array}{c} t_{1, 1} & t_{1, 2} & t_{1, 3} \\ t_{2, 1} & t_{2, 2} & t_{2, 3} \\ t_{3, 1} & t_{3, 2} & t_{3, 3} \end{array} | & = \begin{aligned} t_{1, 1} t_{2, 2} t_{3, 3} | P_{ϕ_{1}} | + t_{1, 1} t_{2, 3} t_{3, 2} | P_{ϕ_{2}} | \\ + t_{1, 2} t_{2, 1} t_{3, 3} | P_{ϕ_{3}} | + t_{1, 2} t_{2, 3} t_{3, 1} | P_{ϕ_{4}} | \\ + t_{1, 3} t_{2, 1} t_{3, 2} | P_{ϕ_{5}} | + t_{1, 3} t_{2, 2} t_{3, 1} | P_{ϕ_{6}} | \end{aligned} \\ = \begin{aligned} t_{1, 1} t_{2, 2} t_{3, 3} (+ 1) + t_{1, 1} t_{2, 3} t_{3, 2} (- 1) \\ + t_{1, 2} t_{2, 1} t_{3, 3} (- 1) + t_{1, 2} t_{2, 3} t_{3, 1} (+ 1) \\ + t_{1, 3} t_{2, 1} t_{3, 2} (+ 1) + t_{1, 3} t_{2, 2} t_{3, 1} (- 1) \end{aligned} \end{aligned}$
Exercise 3.24 Supplied answer

List all of the $4$ -permutations.

Back to Exercise 3.24

Answer. This is all of the permutations where $ϕ (1) = 1$

$\begin{aligned} ϕ_{1} = ⟨ 1, 2, 3, 4 ⟩ ϕ_{2} = ⟨ 1, 2, 4, 3 ⟩ ϕ_{3} = ⟨ 1, 3, 2, 4 ⟩ \\ ϕ_{4} = ⟨ 1, 3, 4, 2 ⟩ ϕ_{5} = ⟨ 1, 4, 2, 3 ⟩ ϕ_{6} = ⟨ 1, 4, 3, 2 ⟩ \end{aligned}$

the ones where $ϕ (1) = 1$

$\begin{aligned} ϕ_{7} = ⟨ 2, 1, 3, 4 ⟩ ϕ_{8} = ⟨ 2, 1, 4, 3 ⟩ ϕ_{9} = ⟨ 2, 3, 1, 4 ⟩ \\ ϕ_{10} = ⟨ 2, 3, 4, 1 ⟩ ϕ_{11} = ⟨ 2, 4, 1, 3 ⟩ ϕ_{12} = ⟨ 2, 4, 3, 1 ⟩ \end{aligned}$

the ones where $ϕ (1) = 3$

$\begin{aligned} ϕ_{13} = ⟨ 3, 1, 2, 4 ⟩ ϕ_{14} = ⟨ 3, 1, 4, 2 ⟩ ϕ_{15} = ⟨ 3, 2, 1, 4 ⟩ \\ ϕ_{16} = ⟨ 3, 2, 4, 1 ⟩ ϕ_{17} = ⟨ 3, 4, 1, 2 ⟩ ϕ_{18} = ⟨ 3, 4, 2, 1 ⟩ \end{aligned}$

and the ones where $ϕ (1) = 4$ .

$\begin{aligned} ϕ_{19} = ⟨ 4, 1, 2, 3 ⟩ ϕ_{20} = ⟨ 4, 1, 3, 2 ⟩ ϕ_{21} = ⟨ 4, 2, 1, 3 ⟩ \\ ϕ_{22} = ⟨ 4, 2, 3, 1 ⟩ ϕ_{23} = ⟨ 4, 3, 1, 2 ⟩ ϕ_{24} = ⟨ 4, 3, 2, 1 ⟩ \end{aligned}$

Exercise 3.25 Supplied answer

A permutation, regarded as a function from the set ${1, . ., n}$ to itself, is one-to-one and onto. Therefore, each permutation has an inverse.

Find the inverse of each $2$ -permutation.
Find the inverse of each $3$ -permutation.

Back to Exercise 3.25

Answer. Each of these is easy to check.

permutation $ϕ_{1}$ $ϕ_{2}$

inverse $ϕ_{1}$ $ϕ_{2}$

permutation	$ϕ_{1}$	$ϕ_{2}$
inverse	$ϕ_{1}$	$ϕ_{2}$

permutation	$ϕ_{1}$	$ϕ_{2}$	$ϕ_{3}$	$ϕ_{4}$	$ϕ_{5}$	$ϕ_{6}$
inverse	$ϕ_{1}$	$ϕ_{2}$	$ϕ_{3}$	$ϕ_{5}$	$ϕ_{4}$	$ϕ_{6}$

Exercise 3.26 Supplied answer

Prove that $f$ is multilinear if and only if for all $\vec{v}, \vec{w} \in V$ and $k_{1}, k_{2} \in ℝ$ , this holds.

$f ({\vec{ρ}}_{1}, \dots, k_{1} {\vec{v}}_{1} + k_{2} {\vec{v}}_{2}, \dots, {\vec{ρ}}_{n}) = k_{1} f ({\vec{ρ}}_{1}, \dots, {\vec{v}}_{1}, \dots, {\vec{ρ}}_{n}) + k_{2} f ({\vec{ρ}}_{1}, \dots, {\vec{v}}_{2}, \dots, {\vec{ρ}}_{n})$

Back to Exercise 3.26

Answer. For the ‘if’ half, the first condition of Definition 3.2 follows from taking $k_{1} = k_{2} = 1$ and the second condition follows from taking $k_{2} = 0$ .

The ‘only if’ half also routine. From $f ({\vec{ρ}}_{1}, \dots, k_{1} {\vec{v}}_{1} + k_{2} {\vec{v}}_{2}, \dots, {\vec{ρ}}_{n})$ the first condition of Definition 3.2 gives $= f ({\vec{ρ}}_{1}, \dots, k_{1} {\vec{v}}_{1}, \dots, {\vec{ρ}}_{n}) + f ({\vec{ρ}}_{1}, \dots, k_{2} {\vec{v}}_{2}, \dots, {\vec{ρ}}_{n})$ and the second condition, applied twice, gives the result.
Exercise 3.27 Supplied answer

How would determinants change if we changed property (4) of the definition to read that $| I | = 2$ ?

Back to Exercise 3.27

Answer. They would all double.
Exercise 3.28 Supplied answer

Verify the second and third statements in Corollary 3.16.

Back to Exercise 3.28

Answer. For the second statement, given a matrix, transpose it, swap rows, and transpose back. The result is swapped columns, and the determinant changes by a factor of $- 1$ . The third statement is similar: given a matrix, transpose it, apply multilinearity to what are now rows, and then transpose back the resulting matrices.
Exercise 3.29 Supplied answer

Recommended. Show that if an $n \times n$ matrix has a nonzero determinant then we can express any column vector $\vec{v} \in ℝ^{n}$ as a linear combination of the columns of the matrix.

Back to Exercise 3.29

Answer. An $n \times n$ matrix with a nonzero determinant has rank $n$ so its columns form a basis for $ℝ^{n}$ .
Exercise 3.30 Supplied answer

[Strang 80] True or false: a matrix whose entries are only zeros or ones has a determinant equal to zero, one, or negative one.

Back to Exercise 3.30

Answer. False.

$| \begin{matrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{matrix} | = 2$
Exercise 3.31 Supplied answer
1. Show that there are $120$ terms in the permutation expansion formula of a $5 \times 5$ matrix.
2. How many are sure to be zero if the $1, 2$ entry is zero?
Back to Exercise 3.31

Answer.
1. For the column index of the entry in the first row there are five choices. Then, for the column index of the entry in the second row there are four choices. Continuing, we get $5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120$ . (See also the next question.)
2. Once we choose the second column in the first row, we can choose the other entries in $4 \cdot 3 \cdot 2 \cdot 1 = 24$ ways.
Exercise 3.32 Supplied answer

How many $n$ -permutations are there?

Back to Exercise 3.32

Answer. $n \cdot (n - 1) \dots 2 \cdot 1 = n!$
Exercise 3.33 Supplied answer

Show that the inverse of a permutation matrix is its transpose.

Back to Exercise 3.33

Answer. [Schmidt] We will show that $P P^{𝖳} = I$ ; the $P^{𝖳} P = I$ argument is similar. The $i, j$ entry of $P P^{𝖳}$ is the sum of terms of the form $p_{i, k} q_{k, j}$ where the entries of $P^{𝖳}$ are denoted with $q$ ’s, that is, $q_{k, j} = p_{j, k}$ . Thus the $i, j$ entry of $P P^{𝖳}$ is the sum $\sum_{k = 1}^{n} p_{i, k} p_{j, k}$ . But $p_{i, k}$ is usually $0$ , and so $P_{i, k} P_{j, k}$ is usually $0$ . The only time $P_{i, k}$ is nonzero is when it is $1$ , but then there are no other $i^{'} \neq i$ such that $P_{i^{'}, k}$ is nonzero ( $i$ is the only row with a $1$ in column $k$ ). In other words,

$\sum_{k = 1}^{n} p_{i, k} p_{j, k} = {\begin{cases} 1 & i = j \\ 0 & otherwise \end{cases}$

and this is exactly the formula for the entries of the identity matrix.
Exercise 3.34 Supplied answer

A matrix $A$ is skew-symmetric if $A^{𝖳} = - A$ , as in this matrix.

$A = (\begin{matrix} 0 & 3 \\ - 3 & 0 \end{matrix})$

Show that $n \times n$ skew-symmetric matrices with nonzero determinants exist only for even $n$ .

Back to Exercise 3.34

Answer. In $| A | = | A^{𝖳} | = | - A | = (- 1)^{n} | A |$ the exponent $n$ must be even.
Exercise 3.35 Supplied answer

Recommended. What is the smallest number of zeros, and the placement of those zeros, needed to ensure that a $4 \times 4$ matrix has a determinant of zero?

Back to Exercise 3.35

Answer. Showing that no placement of three zeros suffices is routine. Four zeroes does suffice; put them all in the same row or column.
Exercise 3.36 Supplied answer

If we have $n$ data points $(x_{1}, y_{1}), (x_{2}, y_{2}), \dots, (x_{n}, y_{n})$ and want to find a polynomial $p (x) = a_{n - 1} x^{n - 1} + a_{n - 2} x^{n - 2} + \dots + a_{1} x + a_{0}$ passing through those points then we can plug in the points to get an $n$ equation/ $n$ unknown linear system. The matrix of coefficients for that system is the Vandermonde matrix. Prove that the determinant of the transpose of that matrix of coefficients

$| \begin{matrix} 1 & 1 & \dots & 1 \\ x_{1} & x_{2} & \dots & x_{n} \\ {x_{1}}^{2} & {x_{2}}^{2} & \dots & {x_{n}}^{2} \\ ⋮ \\ {x_{1}}^{n - 1} & {x_{2}}^{n - 1} & \dots & {x_{n}}^{n - 1} \end{matrix} |$

equals the product, over all indices $i, j \in {1, \dots, n}$ with $i < j$ , of terms of the form $x_{j} - x_{i}$ . (This shows that the determinant is zero, and the linear system has no solution, if and only if the $x_{i}$ ’s in the data are not distinct.)

Back to Exercise 3.36

Answer. The $n = 3$ case shows what to do. The row combination operations of $- x_{1} ρ_{2} + ρ_{3}$ and $- x_{1} ρ_{1} + ρ_{2}$ give this.

$\begin{matrix} | \begin{matrix} 1 & 1 & 1 \\ x_{1} & x_{2} & x_{3} \\ x_{1}^{2} & x_{2}^{2} & x_{3}^{2} \end{matrix} | = | \begin{matrix} 1 & 1 & 1 \\ x_{1} & x_{2} & x_{3} \\ 0 & (- x_{1} + x_{2}) x_{2} & (- x_{1} + x_{3}) x_{3} \end{matrix} | \\ = | \begin{matrix} 1 & 1 & 1 \\ 0 & - x_{1} + x_{2} & - x_{1} + x_{3} \\ 0 & (- x_{1} + x_{2}) x_{2} & (- x_{1} + x_{3}) x_{3} \end{matrix} | \end{matrix}$

Then the row combination operation of $x_{2} ρ_{2} + ρ_{3}$ gives the desired result.

$= | \begin{matrix} 1 & 1 & 1 \\ 0 & - x_{1} + x_{2} & - x_{1} + x_{3} \\ 0 & 0 & (- x_{1} + x_{3}) (- x_{2} + x_{3}) \end{matrix} | = (x_{2} - x_{1}) (x_{3} - x_{1}) (x_{3} - x_{2})$
Exercise 3.37 Supplied answer

We can divide a matrix into blocks, as here,

$(\begin{array}{rrr} 1 & 2 & 0 \\ 3 & 4 & 0 \\ 0 & 0 & - 2 \end{array})$

which shows four blocks, the square $2 \times 2$ and $1 \times 1$ ones in the upper left and lower right, and the zero blocks in the upper right and lower left. Show that if a matrix is such that we can partition it as

$T = (\begin{array}{cc} J & Z_{2} \\ Z_{1} & K \end{array})$

where $J$ and $K$ are square, and $Z_{1}$ and $Z_{2}$ are all zeroes, then $| T | = | J | \cdot | K |$ .

Back to Exercise 3.37

Answer. Let $T$ be $n \times n$ , let $J$ be $p \times p$ , and let $K$ be $q \times q$ . Apply the permutation expansion formula

$| T | = \sum_{permutations ϕ} t_{1, ϕ (1)} t_{2, ϕ (2)} \dots t_{n, ϕ (n)} | P_{ϕ} |$

Because the upper right of $T$ is all zeroes, if a $ϕ$ has at least one of $p + 1, \dots, n$ among its first $p$ column numbers $ϕ (1), \dots, ϕ (p)$ then the term arising from $ϕ$ is $0$ (e.g., if $ϕ (1) = n$ then $t_{1, ϕ (1)} t_{2, ϕ (2)} \dots t_{n, ϕ (n)}$ is $0$ ). So the above formula reduces to a sum over all permutations with two halves: first rearrange $1, \dots, p$ and after that comes a permutation of $p + 1, \dots, p + q$ . To see this gives $| J | \cdot | K |$ , distribute.

$\begin{matrix} [\sum_{\begin{matrix} perms ϕ_{1} \\ of 1, \dots, p \end{matrix}} t_{1, ϕ_{1} (1)} \dots t_{p, ϕ_{1} (p)} | P_{ϕ_{1}} |] \\ \cdot [\sum_{\begin{matrix} perms ϕ_{2} \\ of p + 1, \dots, p + q \end{matrix}} t_{p + 1, ϕ_{2} (p + 1)} \dots t_{p + q, ϕ_{2} (p + q)} | P_{ϕ_{2}} |] \end{matrix}$
Exercise 3.38 Supplied answer

Prove that for any $n \times n$ matrix $T$ there are at most $n$ distinct reals $r$ such that the matrix $T - r I$ has determinant zero (we shall use this result in Chapter Five).

Back to Exercise 3.38

Answer. The $n = 3$ case shows what happens.

$| T - r I | = | \begin{matrix} t_{1, 1} - x & t_{1, 2} & t_{1, 3} \\ t_{2, 1} & t_{2, 2} - x & t_{2, 3} \\ t_{3, 1} & t_{3, 2} & t_{3, 3} - x \end{matrix} |$

Each term in the permutation expansion has three factors drawn from entries in the matrix (e.g., $(t_{1, 1} - x) (t_{2, 2} - x) (t_{3, 3} - x)$ and $(t_{1, 1} - x) (t_{2, 3}) (t_{3, 2})$ ), and so the determinant is expressible as a polynomial in $x$ of degree $3$ . Such a polynomial has at most $3$ roots.

In general, the permutation expansion shows that the determinant is a sum of terms, each with $n$ factors, giving a polynomial of degree $n$ . A polynomial of degree $n$ has at most $n$ roots.
Exercise 3.39 Supplied answer

Puzzle. [Math. Mag., Jan. 1963, Q307] The nine positive digits can be arranged into $3 \times 3$ arrays in $9!$ ways. Find the sum of the determinants of these arrays.

Back to Exercise 3.39

Answer. This is how the answer was given in the cited source. When two rows of a determinant are interchanged, the sign of the determinant is changed. When the rows of a three-by-three determinant are permuted, $3$ positive and $3$ negative determinants equal in absolute value are obtained. Hence the $9!$ determinants fall into $9! / 6$ groups, each of which sums to zero.
Exercise 3.40 Supplied answer

[Math. Mag., Jan. 1963, Q237] Show that

$| \begin{matrix} x - 2 & x - 3 & x - 4 \\ x + 1 & x - 1 & x - 3 \\ x - 4 & x - 7 & x - 10 \end{matrix} | = 0.$

Back to Exercise 3.40

Answer. This is how the answer was given in the cited source. When the elements of any column are subtracted from the elements of each of the other two, the elements in two of the columns of the derived determinant are proportional, so the determinant vanishes. That is,

$| \begin{matrix} 2 & 1 & x - 4 \\ 4 & 2 & x - 3 \\ 6 & 3 & x - 10 \end{matrix} | = | \begin{matrix} 1 & x - 3 & - 1 \\ 2 & x - 1 & - 2 \\ 3 & x - 7 & - 3 \end{matrix} | = | \begin{matrix} x - 2 & - 1 & - 2 \\ x + 1 & - 2 & - 4 \\ x - 4 & - 3 & - 6 \end{matrix} | = 0.$
Exercise 3.41 Supplied answer

Puzzle. [Am. Math. Mon., Jan. 1949] Let $S$ be the sum of the integer elements of a magic square of order three and let $D$ be the value of the square considered as a determinant. Show that $D / S$ is an integer.

Back to Exercise 3.41

Answer. This is how the answer was given in the cited source. Let

$\begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \end{array}$

have magic sum $N = S / 3$ . Then

$\begin{aligned} N & = (a + e + i) + (d + e + f) + (g + e + c) \\ - (a + d + g) - (c + f + i) = 3 e \end{aligned}$

and $S = 9 e$ . Hence, adding rows and columns,

$D = | \begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix} | = | \begin{matrix} a & b & c \\ d & e & f \\ 3 e & 3 e & 3 e \end{matrix} | = | \begin{matrix} a & b & 3 e \\ d & e & 3 e \\ 3 e & 3 e & 9 e \end{matrix} | = | \begin{matrix} a & b & e \\ d & e & e \\ 1 & 1 & 1 \end{matrix} | S .$
Exercise 3.42 Supplied answer

Puzzle. [Am. Math. Mon., Jun. 1931] Show that the determinant of the $n^{2}$ elements in the upper left corner of the Pascal triangle

$\begin{array}{cccccc} 1 & 1 & 1 & 1 & . & . \\ 1 & 2 & 3 & . & . \\ 1 & 3 & . & . \\ 1 & . & . \\ . \\ . \end{array}$

has the value unity.

Back to Exercise 3.42

Answer. This is how the answer was given in the cited source. Denote by $D_{n}$ the determinant in question and by $a_{i, j}$ the element in the $i$ -th row and $j$ -th column. Then from the law of formation of the elements we have

$a_{i, j} = a_{i, j - 1} + a_{i - 1, j}, a_{1, j} = a_{i, 1} = 1.$

Subtract each row of $D_{n}$ from the row following it, beginning the process with the last pair of rows. After the $n - 1$ subtractions the above equality shows that the element $a_{i, j}$ is replaced by the element $a_{i, j - 1}$ , and all the elements in the first column, except $a_{1, 1} = 1$ , become zeroes. Now subtract each column from the one following it, beginning with the last pair. After this process the element $a_{i, j - 1}$ is replaced by $a_{i - 1, j - 1}$ , as shown in the above relation. The result of the two operations is to replace $a_{i, j}$ by $a_{i - 1, j - 1}$ , and to reduce each element in the first row and in the first column to zero. Hence $D_{n} = D_{n + i}$ and consequently

$D_{n} = D_{n - 1} = D_{n - 2} = \dots = D_{2} = 1.$

Determinants Exist

This subsection contains proofs of two results from the prior subsection. It is optional. We will use the material developed here only in the Jordan Canonical Form subsection, which is also optional.

We wish to show that for any size $n$ , the determinant function on $n \times n$ matrices is well-defined. The prior subsection develops the permutation expansion formula.

$\begin{aligned} | \begin{array}{c} t_{1, 1} & t_{1, 2} & \dots & t_{1, n} \\ t_{2, 1} & t_{2, 2} & \dots & t_{2, n} \\ ⋮ \\ t_{n, 1} & t_{n, 2} & \dots & t_{n, n} \end{array} | & = \begin{matrix} t_{1, ϕ_{1} (1)} t_{2, ϕ_{1} (2)} \dots t_{n, ϕ_{1} (n)} | P_{ϕ_{1}} | \\ + t_{1, ϕ_{2} (1)} t_{2, ϕ_{2} (2)} \dots t_{n, ϕ_{2} (n)} | P_{ϕ_{2}} | \\ ⋮ \\ + t_{1, ϕ_{k} (1)} t_{2, ϕ_{k} (2)} \dots t_{n, ϕ_{k} (n)} | P_{ϕ_{k}} | \end{matrix} \\ = \sum_{permutations ϕ} t_{1, ϕ (1)} t_{2, ϕ (2)} \dots t_{n, ϕ (n)} | P_{ϕ} | \end{aligned}$

This reduces the problem of showing that the determinant is well-defined to only showing that the determinant is well-defined on the set of permutation matrices.

A permutation matrix can be row-swapped to the identity matrix. So one way that we can calculate its determinant is by keeping track of the number of swaps. However, we still must show that the result is well-defined. Recall what the difficulty is: the determinant of

$P_{ϕ} = (\begin{matrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix})$

could be computed with one swap

$P_{ϕ} ⟶_{\begin{array}{l} \end{array}}^{ρ_{1} \leftrightarrow ρ_{2}} (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix})$

or with three.

$P_{ϕ} ⟶_{\begin{array}{l} \end{array}}^{ρ_{3} \leftrightarrow ρ_{1}} ⟶_{\begin{array}{l} \end{array}}^{ρ_{2} \leftrightarrow ρ_{3}} ⟶_{\begin{array}{l} \end{array}}^{ρ_{1} \leftrightarrow ρ_{3}} (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix})$

Both reductions have an odd number of swaps so in this case we figure that $| P_{ϕ} | = - 1$ but if there were some way to do it with an even number of swaps then we would have the determinant giving two different outputs from a single input. Below, Corollary 4.5 proves that this cannot happen— there is no permutation matrix that can be row-swapped to an identity matrix in two ways, one with an even number of swaps and the other with an odd number of swaps.

Definition 4.1 In a permutation $ϕ = ⟨ \dots, k, \dots, j, \dots ⟩$ , elements such that $k > j$ are in an inversion of their natural order. Similarly, in a permutation matrix two rows

$P_{ϕ} = (\begin{matrix} ⋮ \\ ι_{k} \\ ⋮ \\ ι_{j} \\ ⋮ \end{matrix})$

such that $k > j$ are in an inversion.

Example 4.2 This permutation matrix

$(\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) = (\begin{matrix} ι_{1} \\ ι_{3} \\ ι_{2} \\ ι_{4} \end{matrix})$

has a single inversion, that $ι_{3}$ precedes $ι_{2}$ .

Example 4.3 There are three inversions here:

$(\begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix}) = (\begin{matrix} ι_{3} \\ ι_{2} \\ ι_{1} \end{matrix})$

$ι_{3}$ precedes $ι_{1}$ , $ι_{3}$ precedes $ι_{2}$ , and $ι_{2}$ precedes $ι_{1}$ .

Lemma 4.4 A row-swap in a permutation matrix changes the number of inversions from even to odd, or from odd to even.

Proof Consider a swap of rows $j$ and $k$ , where $k > j$ .

If the two rows are adjacent

$P_{ϕ} = (\begin{matrix} ⋮ \\ ι_{ϕ (j)} \\ ι_{ϕ (k)} \\ ⋮ \end{matrix}) ⟶_{\begin{array}{l} \end{array}}^{ρ_{k} \leftrightarrow ρ_{j}} (\begin{matrix} ⋮ \\ ι_{ϕ (k)} \\ ι_{ϕ (j)} \\ ⋮ \end{matrix})$

then since inversions involving rows not in this pair are not affected, the swap changes the total number of inversions by one, either removing or producing one inversion depending on whether $ϕ (j) > ϕ (k)$ or not. Consequently, the total number of inversions changes from odd to even or from even to odd.

If the rows are not adjacent then we can swap them via a sequence of adjacent swaps, first bringing row $k$ up

$(\begin{matrix} ⋮ \\ ι_{ϕ (j)} \\ ι_{ϕ (j + 1)} \\ ι_{ϕ (j + 2)} \\ ⋮ \\ ι_{ϕ (k)} \\ ⋮ \end{matrix}) ⟶_{\begin{array}{l} \end{array}}^{ρ_{k} \leftrightarrow ρ_{k - 1}} ⟶_{\begin{array}{l} \end{array}}^{ρ_{k - 1} \leftrightarrow ρ_{k - 2}} \dots ⟶_{\begin{array}{l} \end{array}}^{ρ_{j + 1} \leftrightarrow ρ_{j}} (\begin{matrix} ⋮ \\ ι_{ϕ (k)} \\ ι_{ϕ (j)} \\ ι_{ϕ (j + 1)} \\ ⋮ \\ ι_{ϕ (k - 1)} \\ ⋮ \end{matrix})$

and then bringing row $j$ down.

$⟶_{\begin{array}{l} \end{array}}^{ρ_{j + 1} \leftrightarrow ρ_{j + 2}} ⟶_{\begin{array}{l} \end{array}}^{ρ_{j + 2} \leftrightarrow ρ_{j + 3}} \dots ⟶_{\begin{array}{l} \end{array}}^{ρ_{k - 1} \leftrightarrow ρ_{k}} (\begin{matrix} ⋮ \\ ι_{ϕ (k)} \\ ι_{ϕ (j + 1)} \\ ι_{ϕ (j + 2)} \\ ⋮ \\ ι_{ϕ (j)} \\ ⋮ \end{matrix})$

Each of these adjacent swaps changes the number of inversions from odd to even or from even to odd. The total number of swaps $(k - j) + (k - j - 1)$ is odd. Thus, in aggregate, the number of inversions changes from even to odd, or from odd to even.

QED

Corollary 4.5 If a permutation matrix has an odd number of inversions then swapping it to the identity takes an odd number of swaps. If it has an even number of inversions then swapping to the identity takes an even number.

Proof The identity matrix has zero inversions. To change an odd number to zero requires an odd number of swaps, and to change an even number to zero requires an even number of swaps.

QED

Example 4.6 The matrix in Example 4.3 can be brought to the identity with one swap $ρ_{1} \leftrightarrow ρ_{3}$ . (So the number of swaps needn’t be the same as the number of inversions, but the oddness or evenness of the two numbers is the same.)

Definition 4.7 The signum of a permutation $sgn (ϕ)$ is $- 1$ if the number of inversions in $ϕ$ is odd and is $+ 1$ if the number of inversions is even.

Example 4.8 Using the notation for the $3$ -permutations from Example 3.8 we have

$P_{ϕ_{1}} = (\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) P_{ϕ_{2}} = (\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix})$

so $sgn (ϕ_{1}) = 1$ because there are no inversions, while $sgn (ϕ_{2}) = - 1$ because there is one.

We still have not shown that the determinant function is well-defined because we have not considered row operations on permutation matrices other than row swaps. We will finesse this issue. Define a function $d : ℳ_{n \times n} \to ℝ$ by altering the permutation expansion formula, replacing $| P_{ϕ} |$ with $sgn (ϕ)$ .

$d (T) = \sum_{permutations ϕ} t_{1, ϕ (1)} t_{2, ϕ (2)} \dots t_{n, ϕ (n)} \cdot sgn (ϕ)$

The advantage of this formula is that the number of inversions is clearly well-defined—just count them. Therefore, we will be finished showing that an $n \times n$ determinant function exists when we show that $d$ satisfies the conditions in the definition of a determinant.

Lemma 4.9 The function $d$ above is a determinant. Hence determinant functions $\underset{n \times n}{det}$ exist for every $n$ .

Proof We must check that it satisfies the four conditions from the definition of determinant, Definition 2.1.

Condition (4) is easy: where $I$ is the $n \times n$ identity, in

$d (I) = \sum_{perm ϕ} ι_{1, ϕ (1)} ι_{2, ϕ (2)} \dots ι_{n, ϕ (n)} sgn (ϕ)$

all of the terms in the summation are zero except for the one where the permutation $ϕ$ is the identity, which gives the product down the diagonal, which is one.

For condition (3) suppose that $T ⟶_{\begin{array}{l} \end{array}}^{k ρ_{i}} \hat{T}$ and consider $d (\hat{T})$ .

$\begin{matrix} \sum_{perm ϕ} {\hat{t}}_{1, ϕ (1)} \dots {\hat{t}}_{i, ϕ (i)} \dots {\hat{t}}_{n, ϕ (n)} sgn (ϕ) \\ = \sum_{ϕ} t_{1, ϕ (1)} \dots k t_{i, ϕ (i)} \dots t_{n, ϕ (n)} sgn (ϕ) \end{matrix}$

Factor out $k$ to get the desired equality.

$= k \cdot \sum_{ϕ} t_{1, ϕ (1)} \dots t_{i, ϕ (i)} \dots t_{n, ϕ (n)} sgn (ϕ) = k \cdot d (T)$

For (2) suppose that $T ⟶_{\begin{array}{l} \end{array}}^{ρ_{i} \leftrightarrow ρ_{j}} \hat{T}$ . We must show that $d (\hat{T})$ is the negative of $d (T)$ .

$\begin{matrix} d (\hat{T}) = \sum_{perm ϕ} {\hat{t}}_{1, ϕ (1)} \dots {\hat{t}}_{i, ϕ (i)} \dots {\hat{t}}_{j, ϕ (j)} \dots {\hat{t}}_{n, ϕ (n)} sgn (ϕ) & (*) \end{matrix}$

We will show that each term in ( $*$ ) is associated with a term in $d (T)$ , and that the two terms are negatives of each other. Consider the matrix from the multilinear expansion of $d (\hat{T})$ giving the term ${\hat{t}}_{1, ϕ (1)} \dots {\hat{t}}_{i, ϕ (i)} \dots {\hat{t}}_{j, ϕ (j)} \dots {\hat{t}}_{n, ϕ (n)} sgn (ϕ)$ .

$(\begin{matrix} ⋮ \\ {\hat{t}}_{i, ϕ (i)} \\ ⋮ \\ {\hat{t}}_{j, ϕ (j)} \\ ⋮ \end{matrix})$

It is the result of the $ρ_{i} \leftrightarrow ρ_{j}$ operation performed on this matrix.

$(\begin{matrix} ⋮ \\ t_{i, ϕ (j)} \\ ⋮ \\ t_{j, ϕ (i)} \\ ⋮ \end{matrix})$

That is, the term with hatted $t$ ’s is associated with this term from the $d (T)$ expansion: $t_{1, σ (1)} \dots t_{j, σ (j)} \dots t_{i, σ (i)} \dots t_{n, σ (n)} sgn (σ)$ , where the permutation $σ$ equals $ϕ$ but with the $i$ -th and $j$ -th numbers interchanged, $σ (i) = ϕ (j)$ and $σ (j) = ϕ (i)$ . The two terms have the same multiplicands ${\hat{t}}_{1, ϕ (1)} = t_{1, σ (1)}$ , …, including the entries from the swapped rows ${\hat{t}}_{i, ϕ (i)} = t_{j, ϕ (i)} = t_{j, σ (j)}$ and ${\hat{t}}_{j, ϕ (j)} = t_{i, ϕ (j)} = t_{i, σ (i)}$ . But the two terms are negatives of each other since $sgn (ϕ) = - sgn (σ)$ by Lemma 4.4.

Now, any permutation $ϕ$ can be derived from some other permutation $σ$ by such a swap, in one and only one way. Therefore the summation in ( $*$ ) is in fact a sum over all permutations, taken once and only once.

$\begin{aligned} d (\hat{T}) & = \sum_{perm ϕ} {\hat{t}}_{1, ϕ (1)} \dots {\hat{t}}_{i, ϕ (i)} \dots {\hat{t}}_{j, ϕ (j)} \dots {\hat{t}}_{n, ϕ (n)} sgn (ϕ) \\ = \sum_{perm σ} t_{1, σ (1)} \dots t_{j, σ (j)} \dots t_{i, σ (i)} \dots t_{n, σ (n)} \cdot (- sgn (σ)) \end{aligned}$

Thus $d (\hat{T}) = - d (T)$ .

Finally, for condition (1) suppose that $T ⟶_{\begin{array}{l} \end{array}}^{k ρ_{i} + ρ_{j}} \hat{T}$ .

$\begin{aligned} d (\hat{T}) & = \sum_{perm ϕ} {\hat{t}}_{1, ϕ (1)} \dots {\hat{t}}_{i, ϕ (i)} \dots {\hat{t}}_{j, ϕ (j)} \dots {\hat{t}}_{n, ϕ (n)} sgn (ϕ) \\ = \sum_{ϕ} t_{1, ϕ (1)} \dots t_{i, ϕ (i)} \dots (k t_{i, ϕ (j)} + t_{j, ϕ (j)}) \dots t_{n, ϕ (n)} sgn (ϕ) \end{aligned}$

Distribute over the addition in $k t_{i, ϕ (j)} + t_{j, ϕ (j)}$ .

$= \begin{aligned} \sum_{ϕ} [t_{1, ϕ (1)} \dots t_{i, ϕ (i)} \dots k t_{i, ϕ (j)} \dots t_{n, ϕ (n)} sgn (ϕ) \\ + t_{1, ϕ (1)} \dots t_{i, ϕ (i)} \dots t_{j, ϕ (j)} \dots t_{n, ϕ (n)} sgn (ϕ)] \end{aligned}$

Break it into two summations.

$= \begin{aligned} \sum_{ϕ} t_{1, ϕ (1)} \dots t_{i, ϕ (i)} \dots k t_{i, ϕ (j)} \dots t_{n, ϕ (n)} sgn (ϕ) \\ + \sum_{ϕ} t_{1, ϕ (1)} \dots t_{i, ϕ (i)} \dots t_{j, ϕ (j)} \dots t_{n, ϕ (n)} sgn (ϕ) \end{aligned}$

Recognize the second one.

$= \begin{aligned} k \cdot \sum_{ϕ} t_{1, ϕ (1)} \dots t_{i, ϕ (i)} \dots t_{i, ϕ (j)} \dots t_{n, ϕ (n)} sgn (ϕ) \\ + d (T) \end{aligned}$

Consider the terms $t_{1, ϕ (1)} \dots t_{i, ϕ (i)} \dots t_{i, ϕ (j)} \dots t_{n, ϕ (n)} sgn (ϕ)$ . Notice the subscripts; the entry is $t_{i, ϕ (j)}$ , not $t_{j, ϕ (j)}$ . The sum of these terms is the determinant of a matrix $S$ that is equal to $T$ except that row $j$ of $S$ is a copy of row $i$ of $T$ , that is, $S$ has two equal rows. In the same way that we proved Lemma 2.4 we can see that $d (S) = 0$ : a swap of $S$ ’s equal rows will change the sign of $d (S)$ but since the matrix is unchanged by that swap the value of $d (S)$ must also be unchanged, and so that value must be zero.

QED

We have now proved that determinant functions exist for each size $n \times n$ . We already know that for each size there is at most one determinant. Therefore, for each size there is one and only one determinant function.

We end this subsection by proving the other result remaining from the prior subsection.

Theorem 4.10 The determinant of a matrix equals the determinant of its transpose.

Proof The proof is best understood by doing the general $3 \times 3$ case. That the argument applies to the $n \times n$ case will be clear.

Compare the permutation expansion of the matrix $T$

$\begin{aligned} | \begin{array}{c} t_{1, 1} & t_{1, 2} & t_{1, 3} \\ t_{2, 1} & t_{2, 2} & t_{2, 3} \\ t_{3, 1} & t_{3, 2} & t_{3, 3} \end{array} | & = t_{1, 1} t_{2, 2} t_{3, 3} | \begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} | + t_{1, 1} t_{2, 3} t_{3, 2} | \begin{array}{c} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array} | \\ + t_{1, 2} t_{2, 1} t_{3, 3} | \begin{array}{c} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array} | + t_{1, 2} t_{2, 3} t_{3, 1} | \begin{array}{c} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array} | \\ + t_{1, 3} t_{2, 1} t_{3, 2} | \begin{array}{c} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array} | + t_{1, 3} t_{2, 2} t_{3, 1} | \begin{array}{c} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} | \end{aligned}$

with the permutation expansion of its transpose.

$\begin{aligned} | \begin{array}{c} t_{1, 1} & t_{2, 1} & t_{3, 1} \\ t_{1, 2} & t_{2, 2} & t_{3, 2} \\ t_{1, 3} & t_{2, 3} & t_{3, 3} \end{array} | & = t_{1, 1} t_{2, 2} t_{3, 3} | \begin{array}{c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} | + t_{1, 1} t_{3, 2} t_{2, 3} | \begin{array}{c} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array} | \\ + t_{2, 1} t_{1, 2} t_{3, 3} | \begin{array}{c} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array} | + t_{2, 1} t_{3, 2} t_{1, 3} | \begin{array}{c} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array} | \\ + t_{3, 1} t_{1, 2} t_{2, 3} | \begin{array}{c} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array} | + t_{3, 1} t_{2, 2} t_{1, 3} | \begin{array}{c} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} | \end{aligned}$

Compare first the six products of $t$ ’s. The ones in the expansion of $T$ are the same as the ones in the expansion of the transpose; for instance, $t_{1, 2} t_{2, 3} t_{3, 1}$ is in the top and $t_{3, 1} t_{1, 2} t_{2, 3}$ is in the bottom. That’s perfectly sensible—the six in the top arise from all of the ways of picking one entry of $T$ from each row and column while the six in the bottom are all of the ways of picking one entry of $T$ from each column and row, so of course they are the same set.

Next observe that in the two expansions, each $t$ -product expression is not necessarily associated with the same permutation matrix. For instance, on the top $t_{1, 2} t_{2, 3} t_{3, 1}$ is associated with the matrix for the map $1 \mapsto 2$ , $2 \mapsto 3$ , $3 \mapsto 1$ . On the bottom $t_{3, 1} t_{1, 2} t_{2, 3}$ is associated with the matrix for the map $1 \mapsto 3$ , $2 \mapsto 1$ , $3 \mapsto 2$ . The second map is inverse to the first. This is also perfectly sensible—both the matrix transpose and the map inverse flip the $1, 2$ to $2, 1$ , flip the $2, 3$ to $3, 2$ , and flip $3, 1$ to $1, 3$ .

We finish by noting that the determinant of $P_{ϕ}$ equals the determinant of $P_{ϕ^{- 1}}$ , as Exercise 4.16 shows.

QED

Exercises

These summarize the notation used in this book for the $2$ - and $3$ -permutations.

$i$	$1$	$2$
$ϕ_{1} (i)$	$1$	$2$
$ϕ_{2} (i)$	$2$	$1$

$i$	$1$	$2$	$3$
$ϕ_{1} (i)$	$1$	$2$	$3$
$ϕ_{2} (i)$	$1$	$3$	$2$
$ϕ_{3} (i)$	$2$	$1$	$3$
$ϕ_{4} (i)$	$2$	$3$	$1$
$ϕ_{5} (i)$	$3$	$1$	$2$
$ϕ_{6} (i)$	$3$	$2$	$1$

Exercise 4.11 Supplied answer

Give the permutation expansion of a general $2 \times 2$ matrix and its transpose.

Back to Exercise 4.11

Answer. This is the permutation expansion of the determinant of a $2 \times 2$ matrix

$| \begin{matrix} a & b \\ c & d \end{matrix} | = a d \cdot | \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} | + b c \cdot | \begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix} |$

and the permutation expansion of the determinant of its transpose.

$| \begin{matrix} a & c \\ b & d \end{matrix} | = a d \cdot | \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} | + c b \cdot | \begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix} |$

As with the $3 \times 3$ expansions described in the subsection, the permutation matrices from corresponding terms are transposes (although this is disguised by the fact that each is self-transpose).

Exercise 4.12 Supplied answer

Recommended. This problem appears also in the prior subsection.

Find the inverse of each $2$ -permutation.
Find the inverse of each $3$ -permutation.

Back to Exercise 4.12

Answer. Each of these is easy to check.

permutation $ϕ_{1}$ $ϕ_{2}$

inverse $ϕ_{1}$ $ϕ_{2}$

permutation	$ϕ_{1}$	$ϕ_{2}$
inverse	$ϕ_{1}$	$ϕ_{2}$

permutation	$ϕ_{1}$	$ϕ_{2}$	$ϕ_{3}$	$ϕ_{4}$	$ϕ_{5}$	$ϕ_{6}$
inverse	$ϕ_{1}$	$ϕ_{2}$	$ϕ_{3}$	$ϕ_{5}$	$ϕ_{4}$	$ϕ_{6}$

Exercise 4.13 Supplied answer

Recommended.
1. Find the signum of each $2$ -permutation.
2. Find the signum of each $3$ -permutation.
Back to Exercise 4.13

Answer.
1. $sgn (ϕ_{1}) = + 1$ , $sgn (ϕ_{2}) = - 1$
2. $sgn (ϕ_{1}) = + 1$ , $sgn (ϕ_{2}) = - 1$ , $sgn (ϕ_{3}) = - 1$ , $sgn (ϕ_{4}) = + 1$ , $sgn (ϕ_{5}) = + 1$ , $sgn (ϕ_{6}) = - 1$
Exercise 4.14 Supplied answer

Find the only nonzero term in the permutation expansion of this matrix.

$| \begin{matrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{matrix} |$

Compute that determinant by finding the signum of the associated permutation.

Back to Exercise 4.14

Answer. To get a nonzero term in the permutation expansion we must use the $1, 2$ entry and the $4, 3$ entry. Having fixed on those two we must also use the $2, 1$ entry and the $3, 4$ entry. The signum of $⟨ 2, 1, 4, 3 ⟩$ is $+ 1$ because from

$(\begin{matrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{matrix})$

the two row swaps $ρ_{1} \leftrightarrow ρ_{2}$ and $ρ_{3} \leftrightarrow ρ_{4}$ will produce the identity matrix.
Exercise 4.15 Supplied answer

[Strang 80] What is the signum of the $n$ -permutation $ϕ = ⟨ n, n - 1, \dots, 2, 1 ⟩$ ?

Back to Exercise 4.15

Answer. The pattern is this.

$i$ $1$ $2$ $3$ $4$ $5$ $6$ …

$sgn (ϕ_{i})$ $+ 1$ $- 1$ $- 1$ $+ 1$ $+ 1$ $- 1$ …

So to find the signum of $ϕ_{n!}$ , we subtract one $n! - 1$ and look at the remainder on division by four. If the remainder is $1$ or $2$ then the signum is $- 1$ , otherwise it is $+ 1$ . For $n > 4$ , the number $n!$ is divisible by four, so $n! - 1$ leaves a remainder of $- 1$ on division by four (more properly said, a remainder or $3$ ), and so the signum is $+ 1$ . The $n = 1$ case has a signum of $+ 1$ , the $n = 2$ case has a signum of $- 1$ and the $n = 3$ case has a signum of $- 1$ .
Exercise 4.16 Supplied answer

Prove these.
1. Every permutation has an inverse.
2. $sgn (ϕ^{- 1}) = sgn (ϕ)$
3. Every permutation is the inverse of another.
Back to Exercise 4.16

Answer.
1. We can view permutations as maps $ϕ : {1, \dots, n} \to {1, \dots, n}$ that are one-to-one and onto. Any one-one and onto map has an inverse.
2. If it always takes an odd number of swaps to get from $P_{ϕ}$ to the identity, then it always takes an odd number of swaps to get from the identity to $P_{ϕ}$ (any swap is reversible).
3. This is the first question again.
Exercise 4.17 Supplied answer

Prove that the matrix of the permutation inverse is the transpose of the matrix of the permutation $P_{ϕ^{- 1}} = {P_{ϕ}}^{𝖳}$ , for any permutation $ϕ$ .

Back to Exercise 4.17

Answer. If $ϕ (i) = j$ then $ϕ^{- 1} (j) = i$ . The result now follows on the observation that $P_{ϕ}$ has a $1$ in entry $i, j$ if and only if $ϕ (i) = j$ , and $P_{ϕ^{- 1}}$ has a $1$ in entry $j, i$ if and only if $ϕ^{- 1} (j) = i$ ,
Exercise 4.18 Supplied answer

Recommended. Show that a permutation matrix with $m$ inversions can be row swapped to the identity in $m$ steps. Contrast this with Corollary 4.5.

Back to Exercise 4.18

Answer. This does not say that $m$ is the least number of swaps to produce an identity, nor does it say that $m$ is the most. It instead says that there is a way to swap to the identity in exactly $m$ steps.

Let $ι_{j}$ be the first row that is inverted with respect to a prior row and let $ι_{k}$ be the first row giving that inversion. We have this interval of rows.

$(\begin{matrix} ⋮ \\ ι_{k} \\ ι_{r_{1}} \\ ⋮ \\ ι_{r_{s}} \\ ι_{j} \\ ⋮ \end{matrix}) j < k < r_{1} < \dots < r_{s}$

Swap.

$(\begin{matrix} ⋮ \\ ι_{j} \\ ι_{r_{1}} \\ ⋮ \\ ι_{r_{s}} \\ ι_{k} \\ ⋮ \end{matrix})$

The second matrix has one fewer inversion because there is one fewer inversion in the interval ( $s$ vs. $s + 1$ ) and inversions involving rows outside the interval are not affected.

Proceed in this way, at each step reducing the number of inversions by one with each row swap. When no inversions remain the result is the identity.

The contrast with Corollary 4.5 is that the statement of this exercise is a ‘there exists’ statement: there exists a way to swap to the identity in exactly $m$ steps. But the corollary is a ‘for all’ statement: for all ways to swap to the identity, the parity (evenness or oddness) is the same.

$i$	$1$	$2$	$3$	$4$	$5$	$6$	…
$sgn (ϕ_{i})$	$+ 1$	$- 1$	$- 1$	$+ 1$	$+ 1$	$- 1$	…

Exercise 4.19 Supplied answer

Recommended. For any permutation $ϕ$ let $g (ϕ)$ be the integer defined in this way.

$g (ϕ) = \prod_{i < j} [ϕ (j) - ϕ (i)]$

(This is the product, over all indices $i$ and $j$ with $i < j$ , of terms of the given form.)

Compute the value of $g$ on all $2$ -permutations.
Compute the value of $g$ on all $3$ -permutations.
Prove that $g (ϕ)$ is not $0$ .
Prove this.

$sgn (ϕ) = \frac{g (ϕ)}{| g (ϕ) |}$

Many authors give this formula as the definition of the signum function.

Back to Exercise 4.19

Answer.

First, $g (ϕ_{1})$ is the product of the single factor $2 - 1$ and so $g (ϕ_{1}) = 1$ . Second, $g (ϕ_{2})$ is the product of the single factor $1 - 2$ and so $g (ϕ_{2}) = - 1$ .

permutation $ϕ$	$ϕ_{1}$	$ϕ_{2}$	$ϕ_{3}$	$ϕ_{4}$	$ϕ_{5}$	$ϕ_{6}$
$g (ϕ)$	$2$	$- 2$	$- 2$	$2$	$2$	$- 2$

It is a product of nonzero terms.
Note that $ϕ (j) - ϕ (i)$ is negative if and only if $ι_{ϕ (j)}$ and $ι_{ϕ (i)}$ are in an inversion of their usual order.

References cited in this section

Am. Math. Mon., Apr. 1955

Vern Haggett (proposer), F. W. Saunders (solver), Elementary problem 1135, American Mathematical Monthly, vol. 62 no. 4 (Apr. 1955), p. 257.

Am. Math. Mon., Feb. 1953

Norman Anning (proposer), C. W. Trigg (solver), Elementary problem 1016, American Mathematical Monthly, vol. 60 no. 2 (Feb. 1953), p. 115.

Kemp

Franklin Kemp Linear Equations, American Mathematical Monthly, volume 89 number 8 (Oct. 1982), p. 608.

Strang 80

Gilbert Strang, Linear Algebra and its Applications, second edition, Harcourt Brace Jovanovich, 1980.

Schmidt

Jack Schmidt, http://math.stackexchange.com/a/98558/12012, 2012-Jan-12.

Math. Mag., Jan. 1963, Q307

C. W. Trigg (proposer). Quickie 307, Mathematics Magazine, volume 36 number 1 (Jan. 1963), p. 77.

Math. Mag., Jan. 1963, Q237

D. L. Silverman (proposer), C. W. Trigg (solver), Quickie 237, Mathematics Magazine, volume 36 number 1 (Jan. 1963).

Am. Math. Mon., Jan. 1949

C. W. Trigg (proposer), R. J. Walker (solver), Elementary problem 813, American Mathematical Monthly, vol. 56 no. 1 (Jan. 1949), p. 33.

Am. Math. Mon., Jun. 1931

C. A. Rupp (proposer), H. T. R. Aude (solver), problem 3468, American Mathematical Monthly, vol. 37 no. 6 (June-July 1931), p. 355.