Question

In the following exercises, evaluate each determinant by expanding by minors along the first row.

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

Short answer

The value of the determinant is$-12.$

Step-by-step solution

Step 1. Given Information

The given matrix is

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

We have to evaluate the determinant by expanding by minors along the first row.

Step 2. Expand by minors along the first row

The first row is $\left[\begin{array}{ccc}-2& 3& -1\end{array}\right]$

Now, let's expand by minors along the first row

$=-2\left|\begin{array}{cc}2& -2\\ 1& -3\end{array}\right|-3\left|\begin{array}{cc}-1& -2\\ 3& -3\end{array}\right|-1\left|\begin{array}{cc}-1& 2\\ 3& 1\end{array}\right|$

Step 3. Evaluate the determinant

By expanding by minors along the first row we get,

$$\begin{gathered} =-2\left|\begin{array}{cc}2& -2\\ 1& -3\end{array}\right|-3\left|\begin{array}{cc}-1& -2\\ 3& -3\end{array}\right|-1\left|\begin{array}{cc}-1& 2\\ 3& 1\end{array}\right| \\ =-2\left[\right(2\left)\right(-3)-(1\left)\right(-2\left)\right]-3\left[\right(-1\left)\right(-3)-(3\left)\right(-2\left)\right]-1\left[\right(-1\left)\right(1)-(3\left)\right(2\left)\right] \\ =-2[-4]-3\left[9\right]-1[-7] \\ =8-27+7 \\ =-12 \end{gathered}$$

Thus, the determinant is$-12.$