Linear Algebra with Applications, 10th Edition

2.2 Properties of Determinants

In this section, we consider the effects of row operations on the determinant of a matrix. Once these effects have been established, we will prove that a matrix A is singular if and only if its determinant is zero, and we will develop a method for evaluating determinants by using row operations. Also, we will establish an important theorem about the determinant of the product of two matrices. We begin with the following lemma:

Lemma 2.2.1

Let A be an $n\times n$ matrix. If $A_{\mathit{\text{jk}}}$ denotes the cofactor of $a_{\mathit{\text{jk}}}$ for $k=\mathrm{1, }\mathrm{\dots , }n$, then

$$a_{i1}{A}_{j1}+a_{i2}{A}_{j2}+\cdots +a_{in}{A}_{jn}=\{\begin{array}{cc}\text{det}(A)\hfill & \mathit{\text{if }}i=j\\ 0\hfill & \mathit{\text{if }}i\ne j\end{array}$$ (1)

Proof

If $i=j$, (1) is just the cofactor expansion of det(A) along the ith row of A. To prove (1) in the case $i\ne j$, let $A^{\ast }$ be the matrix obtained by replacing ...