Chapter 7.7, Problem 82E

To determine

To explain the reason for the determinant of the matrix to be equal to zero.

Answer to Problem 82E

The determinant of the matrix is equal to zero because the cofactor for that row would be zero times its cofactors.

Explanation of Solution

Given information :

The determinant of the matrix is given as,

  $\left[\begin{array}{ccc}2& -4& 5\\ 1& -2& 3\\ 0& 0& 0\end{array}\right]$

For a square matrix that has an entire row of zeros, the determinant will always be zero because when expanding by cofactors, the cofactor for that row would be zero times its cofactors that would result the value to be zero.

Therefore,

The determinant of the matrix is equal to zero because the cofactor for that row would be zero times its cofactors.