Question

Evaluate each determinant by expanding by minors.

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

Short answer

The required value of the determinant is$-24$

Step-by-step solution

Step 1. Given information

The determinant is$\left|\begin{array}{ccc}4& -1& 3\\ 3& -2& 2\\ -1& 0& 4\end{array}\right|$

Step 2. Expand by minors along the third row to evaluate determinant

$$\begin{gathered} \left|\begin{array}{ccc}4& -1& 3\\ 3& -2& 2\\ -1& 0& 4\end{array}\right| \\ =-1\left|\begin{array}{cc}-1& 3\\ -2& 2\end{array}\right|-0\left|\begin{array}{cc}4& 3\\ 3& 2\end{array}\right|+4\left|\begin{array}{cc}4& -1\\ 3& -2\end{array}\right| \\ =-1\left[\right(-1\left)\right(2)-(3\left)\right(-2\left)\right]-0+4\left[\right(4\left)\right(-2)-(-1\left)\right(3\left)\right] \\ =-1\left[4\right]-0+4[-5] \\ =-4-0-20 \\ =-24 \end{gathered}$$

Step 3. Required answer

The required value of the determinant is$-24$