Example Find the determinant of the following matrix.

$$\det \left(\begin{array}{rrr} 0&-2&0\\ -1&3&1\\ 4&2&1 \end{array}\right)$$

Answer Suppose that a matrix $A = (a_{ij})$ is $n$th order square matrix．Then the cofactor of $a_{ij}$ is defined as follows:

$$A_{ij} = (-1)^{i+j}M_{ij}.$$

Here $M_{ij}$ is called the minor which can be obtained by eliminating $i$th row and $j$th column. Now the determinant of the matrix $A$ is defined as follows:

$$\det{A} = \sum_{j=1}^{n}a_{ij}A_{ij}.$$

We apply the cofactor expansion using the 1st row. Let $A_{ij}$ be the cofactor of the matrix $A$

$$\det(A_{ij}) = (-1)^{i+j}\det(M_{ij})$$

$$\left \vert\begin{array}{rrr} 0&-2&0\\ -1&3&1\\ 4&2&1 \end{arra... ...(-2)\left\vert\begin{array}{rr} -1&1\\ 4&1 \end{array}\right\vert + 0 = -10 .$$