Chapter 16.5, Problem 15WE

To determine

To show: that if all elements in a column or row of a matrix is zero, then $\det A=0$ .

Explanation of Solution

Given information:

Let $A$ be any $2\times 2$ matrix.

Calculation:

The determinant of a $2\times 2$ matrix $A$ is defined as,

  $\begin{array}{c}\det A=\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|\\ =ad-bc\end{array}$

Consider the case when all elements in a column or row of a matrix is zero. Let the first rows containsall 0 elements.

  $\begin{array}{c}\left|\begin{array}{cc}0& 0\\ c& d\end{array}\right|=\left(0\right)\left(d\right)-\left(0\right)\left(c\right)\\ =0-0\\ =0\end{array}$

Thus, the determinant of a $2\times 2$ matrix will be 0 if all elements in a column or row of a matrix is zero.