Question

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

Short answer

The value of the determinant is 37.

Step-by-step solution

Step 1. Given information 

The given determinant is

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

Step 2. Calculation

Expand by minors along the first row

$$\begin{gathered} 3\left[\begin{array}{cc}-1& -2\\ 3& -1\end{array}\right]-(-2)\left[\begin{array}{cc}0& -2\\ 2& -1\end{array}\right]+4\left[\begin{array}{cc}0& -1\\ 2& 3\end{array}\right] \\ =3(1-(-2\times 3\left)\right)+2\left(\right(0\times -1)-(-2\times 2\left)\right)+4(3\times 0-(-1\times 2\left)\right) \\ =3(1+6)+2(0+4)+4(0+2) \\ =3\times 7+2\times 4+4\times 2 \\ =21+8+8 \\ =37 \end{gathered}$$