3.1 Introduction to determinants 行列式介绍

本文为《Linear algebra and its applications》的读书笔记

目录

• Definition

Definition

Recall from Section 2.2 that a $2 \times 2$2×2 matrix is invertible if and only if its determinant is nonzero. To extend this useful fact to larger matrices, we need a definition for the determinant of an $n \times n$n×n matrix. We can discover the definition for the $3 \times 3$3×3 case by watching what happens when an invertible $3 \times 3$3×3 matrix $A$A is row reduced.

Consider $A = [a_{ij}]$A=[aij​] with $a_{11} \neq 0$a11​​=0. If we multiply the second and third rows of $A$A by $a_{11}$a11​ and then subtract appropriate multiples of the first row from the other two rows, we find that $A$A is row equivalent to the following two matrices:

Since $A$A is invertible, either the $(2, 2)$(2,2)-entry or the $(3, 2)$(3,2)-entry on the right in (1) is nonzero. Let us suppose that the $(2, 2)$(2,2)-entry is nonzero. (Otherwise, we can make a row interchange before proceeding.) Multiply row 3 by $a_{11}a_{22} - a_{12}a_{21}$a11​a22​−a12​a21​, and then to the new row 3 add $-(a_{11}a_{32}-a_{12}a_{31})$−(a11​a32​−a12​a31​) times row 2. This will show that

where
$\Delta=a_{11}a_{22}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{11}a_{23}a_{32}-a_{12}a_{21}a_{33}-a_{13}a_{22}a_{31}\ \ \ \ \ \ \ (2)$Δ=a11​a22​a33​+a12​a23​a31​+a13​a21​a32​−a11​a23​a32​−a12​a21​a33​−a13​a22​a31​       (2)Since $A$A is invertible, $\Delta$Δ must be nonzero. The converse is true, too. We call $\Delta$Δ in (2) the determinant of the $3 \times 3$3×3 matrix $A$A.

Recall that the determinant of a $2 \times 2$2×2 matrix, $A = [a_{ij}]$A=[aij​], is the number
$det\ A=a_{11}a_{22}-a_{12}a_{21}$det A=a11​a22​−a12​a21​For a $1 \times 1$1×1 matrix—say, $A = [a_{11}]$A=[a11​]—we define $det\ A = a_{11}$det A=a11​. To generalize the definition of the determinant to larger matrices, we’ll use $2 \times 2$2×2 determinants to rewrite the $3 \times 3$3×3 determinant $\Delta$Δ described above. Since the terms in $\Delta$Δ can be grouped as $(a_{11}a_{22}a_{33} - a_{11}a_{23}a_{32}) -(a_{12}a_{21}a_{33} - a_{12}a_{23}a_{31})+(a_{13}a_{21}a_{32} - a_{13}a_{22}a_{31})$(a11​a22​a33​−a11​a23​a32​)−(a12​a21​a33​−a12​a23​a31​)+(a13​a21​a32​−a13​a22​a31​).

For brevity, write

where $A_{11}$A11​, $A_{12}$A12​, and $A_{13}$A13​ are obtained from $A$A by deleting the first row and one of the three columns. For any square matrix $A$A, let $A_{ij}$Aij​ denote the submatrix formed by deleting the $i$ith row and $j$jth column of $A$A.

We can now give a $recursive$recursive definition of a determinant.

Another common notation for the determinant of a matrix uses a pair of vertical lines in place of brackets.

To state the next theorem, it is convenient to write the definition of $det\ A$det A in a slightly different form. Given $A = [a_{ij}]$A=[aij​], the $(i, j)$(i,j)-cofactor (余因子) of $A$A is the number $C_{ij}$Cij​ given by

Then

This formula is called a cofactor expansion across the first row (第一行的余因子展开式) of $A$A. We omit the proof of the following fundamental theorem to avoid a lengthy digression.

The factor $(-1)^{i+j}$(−1)i+j determines the following checkerboard pattern of signs:

Theorem 1 is helpful for computing the determinant of a matrix that contains many zeros.

EXAMPLE 3
Compute $det\ A$det A, where

SOLUTION

The matrix in Example 3 was nearly triangular. The method in that example is easily adapted to prove the following theorem.