Question

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

$\left|\begin{array}{ccc}-2& -3& -4\\ 5& -6& 7\\ -1& 2& 0\end{array}\right|$

Short answer

The required value of the determinant is$33$

Step-by-step solution

Step 1. Given information

The determinant is$\left|\begin{array}{ccc}-2& -3& -4\\ 5& -6& 7\\ -1& 2& 0\end{array}\right|$

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

$$\begin{gathered} \left|\begin{array}{ccc}-2& -3& -4\\ 5& -6& 7\\ -1& 2& 0\end{array}\right|=-2\left|\begin{array}{cc}-6& 7\\ 2& 0\end{array}\right|-(-3)\left|\begin{array}{cc}5& 7\\ -1& 0\end{array}\right|-4\left|\begin{array}{cc}5& -6\\ -1& 2\end{array}\right| \\ =-2\left[\right(-6\left)\right(0)-(7\left)\right(2\left)\right]+3\left[\right(5\left)\right(0)-(7\left)\right(-1\left)\right]-4\left[\right(5\left)\right(2)-(-6\left)\right(-1\left)\right] \\ =-2[-14]+3\left[7\right]-4\left[4\right] \\ =28+21-16 \\ =33 \end{gathered}$$

Step 3. Required answer

The required value of the determinant is$33$