Cofactor Expansion

Question

Let $A = [\begin{matrix} 0 & 1 & - 1 \\ 1 & 3 & 2 \\ 4 & 1 & - 2 \end{matrix}]$ . Let $i = 2$ and $j = 3$ .

What is the value of $(- 1)^{i + j} det (A (i ∣ j))$ ?

The answer is 4.

$(- 1)^{i + j} det (A (i ∣ j)) = (- 1)^{2 + 3} det (A (2 ∣ 3)) = - | \begin{array}{cc} 0 & 1 \\ 4 & 1 \end{array} | = - (0 - 4) = 4$ .