Chapter 16.5, Problem 16WE

To determine

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

Explanation of Solution

Given information:

Let $B$ be any $3\times 3$ matrix.

Calculation:

The determinant of a $3\times 3$ matrix $B$ is defined as,

  $\begin{array}{c}\det B=\left|\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right|\\ =a\left|\begin{array}{cc}e& f\\ h& i\end{array}\right|-b\left|\begin{array}{cc}d& f\\ g& i\end{array}\right|+c\left|\begin{array}{cc}d& e\\ g& h\end{array}\right|\\ =a\left(ei-hf\right)-b\left(di-gf\right)+c\left(dh-ge\right)\end{array}$

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

  $\begin{array}{c}\det B=\left|\begin{array}{ccc}0& 0& 0\\ d& e& f\\ g& h& i\end{array}\right|\\ =0\left|\begin{array}{cc}e& f\\ h& i\end{array}\right|-0\left|\begin{array}{cc}d& f\\ g& i\end{array}\right|+0\left|\begin{array}{cc}d& e\\ g& h\end{array}\right|\\ =0\left(ei-hf\right)-0\left(di-gf\right)+0\left(dh-ge\right)\\ =0\end{array}$

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

  $\begin{array}{c}\det B=\left|\begin{array}{ccc}0& b& c\\ 0& e& f\\ 0& h& i\end{array}\right|\\ =0\left|\begin{array}{cc}e& f\\ h& i\end{array}\right|-b\left|\begin{array}{cc}0& f\\ 0& i\end{array}\right|+c\left|\begin{array}{cc}0& e\\ 0& h\end{array}\right|\\ =0\left(ei-hf\right)-b\left(0-0\right)+c\left(0-0\right)\\ =0\end{array}$

Thus, the determinant of a $3\times 3$ matrix will be 0 if all elements in a column or row of a matrix is zero.