Chapter 16.5, Problem 12WE

To determine

To evaluate: the determinant $\left|\begin{array}{ccc}0& 2& -3\\ 3& 5& -3\\ 1& 2& 0\end{array}\right|$ .

Answer to Problem 12WE

  $\left|\begin{array}{ccc}0& 2& -3\\ 3& 5& -3\\ 1& 2& 0\end{array}\right|=-9$

Explanation of Solution

Given:

A determinant as

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

Calculation:

The determinant can be evaluated as,

  $\begin{array}{c}\left|\begin{array}{ccc}0& 2& -3\\ 3& 5& -3\\ 1& 2& 0\end{array}\right|=0\left|\begin{array}{cc}5& -3\\ 2& 0\end{array}\right|-\left(2\right)\left|\begin{array}{cc}3& -3\\ 1& 0\end{array}\right|+\left(-3\right)\left|\begin{array}{cc}3& 5\\ 1& 2\end{array}\right|\\ =0\left[\left(5\right)\left(0\right)-\left(-3\right)\left(2\right)\right]-2\left[\left(0\right)\left(3\right)-\left(1\right)\left(-3\right)\right]-3\left[\left(3\right)\left(2\right)-\left(5\right)\left(1\right)\right]\\ =-2\left[0+3\right]-3\left[6-5\right]\\ =-2\left(3\right)-3\left(1\right)\\ =-6-3\\ =-9\end{array}$