Question

Evaluate each determinant by expanding by minors along the first row.

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

Short answer

The required value of the determinant is$49$

Step-by-step solution

Step 1. Given information

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

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

$$\begin{gathered} \left|\begin{array}{ccc}1& 3& -2\\ 5& -6& 4\\ 0& -2& -1\end{array}\right| \\ =1\left|\begin{array}{cc}-6& 4\\ -2& -1\end{array}\right|-3\left|\begin{array}{cc}5& 4\\ 0& -1\end{array}\right|-2\left|\begin{array}{cc}5& -6\\ 0& -2\end{array}\right| \\ =1\left[\right(-6\left)\right(-1)-(4\left)\right(-2\left)\right]-3\left[\right(5\left)\right(-1)-(4\left)\right(0\left)\right]-2\left[\right(5\left)\right(-2)-(-6\left)\right(0\left)\right] \\ =1\left[14\right]-3[-5]-2[-10] \\ =14+15+20 \\ =49 \end{gathered}$$

Step 3. Required answer

The required value of the determinant is$49$