Chapter 8.4, Problem 94E

To determine

Determine whether the given statement is true or false.

Answer to Problem 94E

The given statement is true

Explanation of Solution

Given:

If two columns of a square matrix is the same, then the determinant will be zero.

If two columns of a square matrix is the same, then the determinant will be zero.

Consider the co-factor expansion of the determinant for any row or column, say row 1. Let column 2 and column 3 be the same.

Then, in the expansion

  $\left|A\right|=a_{11}{C}_{11}+a_{12}{C}_{12}+a_{13}{C}_{13}$

Here, $C_{11}$ will become zero and $a_{22}{C}_{22}$ will be equal to $–a_{23}{C}_{23}$ .

Thus, the determinant becomes zero.