Question

Evaluate each determinant by expanding by minors.

$\left|\begin{array}{ccc}-5& -1& -4\\ 4& 0& -3\\ 2& -2& 6\end{array}\right|$

Short answer

The required value of the determinant is$92$

Step-by-step solution

Step 1. Given information

The determinant is$\left|\begin{array}{ccc}-5& -1& -4\\ 4& 0& -3\\ 2& -2& 6\end{array}\right|$

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

$$\begin{gathered} \left|\begin{array}{ccc}-5& -1& -4\\ 4& 0& -3\\ 2& -2& 6\end{array}\right| \\ =-5\left|\begin{array}{cc}0& -3\\ -2& 6\end{array}\right|-(-1)\left|\begin{array}{cc}4& -3\\ 2& 6\end{array}\right|-4\left|\begin{array}{cc}4& 0\\ 2& -2\end{array}\right| \\ =-5\left[\right(0\left)\right(6)-(-3\left)\right(-2\left)\right]-(-1)\left[\right(4\left)\right(6)-(-3\left)\right(2\left)\right]-4\left[\right(4\left)\right(-2)-(0\left)\right(2\left)\right] \\ =-5[-6]+1\left[30\right]-4[-8] \\ =30+30+32 \\ =92 \end{gathered}$$

Step 3. Required information

The required value of the determinant is$92$