Question

Evaluate the determinant $\left[\begin{array}{ccc}3& -2& -2\\ 2& -1& 4\\ -1& 0& -3\end{array}\right]$, by expanding by minors along the first row.

Short answer

The value of the determinant is 7

Step-by-step solution

Step 1. Given information

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

Step 2. Calculation

Expand by minors along the first row

$$\begin{gathered} 3\left[\begin{array}{cc}-1& 4\\ 0& -3\end{array}\right]-(-2)\left[\begin{array}{cc}2& 4\\ -1& -3\end{array}\right]-2\left[\begin{array}{cc}2& -1\\ -1& 0\end{array}\right] \\ =3(3-(4\times 0\left)\right)+2\left(\right(2\times -3)-(-1\times 4\left)\right)-2(2\times 0-(-1\times -1\left)\right) \\ =3\left[3]+[2(-6+4)]-[2(0-1)\right] \\ =3\left[3]-[4]+[2\right] \\ =9-4+2 \\ =7 \end{gathered}$$