Cofactor Matrix

Definition: Cofactor Matrix

Let $A\in F^{n\times n}$ be a square matrix.

The cofactor matrix of an element $a_{ij}$ of $A$ is the matrix $A_{ij}\in F^{(n-1)\times (n-1)}$ obtained by removing the $i$-th row and the $j$-th column of $A$ and then sticking the rest of its rows and columns together.

EXAMPLE

$$\left[\begin{array}{ccc}6& 7& 8\\ 3& 2& 1\\ 7& 6& 5\end{array}\right]$$

The cofactor matrix of the element $a_{1,1}=6$ is

$$\left[\begin{array}{cc}2& 1\\ 6& 5\end{array}\right]$$

The cofactor matrix of the element $a_{1,2}=7$ is

$$\left[\begin{array}{cc}3& 1\\ 7& 5\end{array}\right]$$

The cofactor matrix of the element $a_{1,3}=8$ is

$$\left[\begin{array}{cc}3& 2\\ 7& 6\end{array}\right]$$