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APPENDIX B Theory of Determinants

The proof of the theorem on row and column cofactor expansions (Theorem 1 in Section 3.6) involves an alternative interpretation of determinants that in more advanced treatments is usually taken as the original definition. Consider the following elementary scheme for evaluating a $3 \times 3$ $3 \times 3$ determinant:

The first two columns of $A = [a_{i j}]$ $A = [a_{i j}]$ are duplicated to its right. It is readily verified that det A is equal to the sum of the three products along the indicated diagonals that point down and to the right, minus the sum of the products along the three diagonals that point down and to the left:

\begin{array}{rcl} \det A & = & + a_{11} a_{22} a_{33} + a_{12} a_{23} a_{31} + a_{13} a_{21} a_{32} \\ - a_{13} a_{22} a_{31} - a_{11} a_{23} a_{32} - a_{12} a_{21} a_{33} . \end{array}

$\begin{array}{rcl} \det A & = & + a_{11} a_{22} a_{33} + a_{12} a_{23} a_{31} + a_{13} a_{21} a_{32} \\ - a_{13} a_{22} a_{31} - a_{11} a_{23} a_{32} - a_{12} a_{21} a_{33} . \end{array}$ (1)

Remark

This scheme works only in dimensions 2 and 3. Neither the analogous scheme nor anything resembling it gives the correct value of an $n \times n$ $n \times n$ determinant with $n \geq 4.$ $n \geq 4.$

Note that each of the six terms on the right in (1) is of the form $\pm a_{1 i} a_{2 j} a_{3 k},$ $\pm a_{1 i} a_{2 j} a_{3 k},$ where $(i j k)$ $(i j k)$ is a permutation of $(1 2 3) .$ $(1 2 3) .$ That is, the triple $(i j k)$ $(i j k)$ consists of the three distinct numbers 1, 2, and 3 written in some specific order. The six terms in (1) correspond to the six possible permutations of $(1 2 3) :$ $(1 2 3) :$

\begin{array}{l} (1 2 3) & (2 1 3) & (3 1 2) \\ (1 3 2) & (2 3 1) & (3 2 1) . \end{array}

$\begin{array}{l} (1 2 3) & (2 1 3) & (3 1 2) \\ (1 3 2) & (2 3 1) & (3 2 1) . \end{array}$ (2)

The $+$ $+$ and $-$ $-$ signs in (1) can also be explained in terms of these permutations. A transposition of an ordered sequence of objects (such as the numbers 1, 2, 3) is the operation of interchanging some single pair of them. For instance, the operation $(1 2 3) \to (1 3 2)$ $(1 2 3) \to (1 3 2)$ is a transposition that consists of interchanging 2 and 3. But it requires two transpositions to change $(1 2 3)$ $(1 2 3)$ into the permutation $(3 1 2) :$ $(3 1 2) :$

(1 2 3) \to (3 2 1) \to (3 1 2) .

$(1 2 3) \to (3 2 1) \to (3 1 2) .$

Given a permutation $P = (i j k)$ $P = (i j k)$ of $(1 2 3),$ $(1 2 3),$ we denote by s(P) the minimum number of transpositions required to change $(1 2 3)$ $(1 2 3)$ into $(i j k) .$ $(i j k) .$ For the six permutations in (2) it is easy to verify that

\begin{array}{l} s (1 2 3) = 0, & s (2 1 3) = 1, & s (3 1 2) = 2, \\ s (1 3 2) = 1, & s (2 3 1) = 2, & s (3 2 1) = 1. \end{array}

$\begin{array}{l} s (1 2 3) = 0, & s (2 1 3) = 1, & s (3 1 2) = 2, \\ s (1 3 2) = 1, & s (2 3 1) = 2, & s (3 2 1) = 1. \end{array}$ (3)

Now for the point to all this: If you check each of the six terms in (1) you will find that the sign of the term $a_{1 i} a_{2 j} a_{3 k}$ $a_{1 i} a_{2 j} a_{3 k}$ is ${(- 1)}^{s (P)},$ ${(- 1)}^{s (P)},$ where $P = (i j k) .$ $P = (i j k) .$ Therefore, we can rewrite (1) more concisely as

det A = \sum_{P} {(- 1)}^{s (P)} a_{1 i} a_{2 j} a_{3 k}

$det A = \sum_{P} {(- 1)}^{s (P)} a_{1 i} a_{2 j} a_{3 k}$ (4)

where there is one term on the right for each possible permutation $P = (i j k)$ $P = (i j k)$ of $(1 2 3) .$ $(1 2 3) .$

The formula in (4) generalizes to determinants of higher order. If A is an $n \times n$ $n \times n$ matrix, then

det A = \sum_{P} {(- 1)}^{s (P)} a_{1 i_{1}} a_{2 i_{2}} \dots a_{n i_{n}},

$det A = \sum_{P} {(- 1)}^{s (P)} a_{1 i_{1}} a_{2 i_{2}} \dots a_{n i_{n}},$ (5)

where there is one term on the right for each possible permutation $P = (i_{1} i_{2} \dots i_{n})$ $P = (i_{1} i_{2} \dots i_{n})$ of $(1 2 \dots n),$ $(1 2 \dots n),$ and s(P) denotes the minimum number of transpositions required to change $(1 2 \dots n)$ $(1 2 \dots n)$ into $(i_{1} i_{2} \dots i_{n}) .$ $(i_{1} i_{2} \dots i_{n}) .$ The formula in (5) can be established by induction on n. Assuming its validity for $(n - 1) \times (n - 1)$ $(n - 1) \times (n - 1)$ determinants, the formula can be verified for an $n \times n$ $n \times n$ matrix A by expanding along its first row. It is fairly easy to see that this gives terms of the form $\pm a_{1 i_{1}} a_{2 i_{2}} \dots a_{n i_{n}},$ $\pm a_{1 i_{1}} a_{2 i_{2}} \dots a_{n i_{n}},$ but somewhat more difficult to check the signs.

Finally, the proof of the cofactor expansion theorem consists of verifying similarly that the cofactor expansion of det A along any row or column of A agrees with the formula in (5). The details are more lengthy than instructive, and therefore we omit them.

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Table of Contents for APPENDIX B Theory of Determinants

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Table of Contents for
APPENDIX B Theory of Determinants