Example Find the determinant of the following matrix.

$\displaystyle \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:

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

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:

$\displaystyle \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 $A$

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

$\displaystyle \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 .
$