# Cofactor Matrix (examples)

Example: $2 \times 2$ matrix
Let $A$ be a $2 \times 2$ matrix defined as
The cofactor matrix of $A$ is

Let $M_{ij}$ be a submatrix given by removing $i$-th row and $j$-th column from $A$,
The determinant of $M_{ij}$ is
Each element of the cofactor matrix is defined as
Specifically, we see that
and in the form of a matrix,
Example
Let $A$ be a $2 \times 2$ matrix given as
Each element of $\tilde{A}$ is
Calculator
Enter a $2 \times 2$ matrix and press "Execute" button. Its cofactor matrix is displayed.

Matrix $A$
 1 2 1 2

Cofactor Matrix $\tilde{A}$
 1 2 1 2

Example: $3 \times 3$ matrix
Let $A$ be a $3 \times 3$ matrix given as
. The cofactor matrix of $A$ is

Let $M_{ij}$ be a submatrix given by removing $i$-th row and $j$-th column from $A$,
The determinant of $M_{ij}$ is
Each element of the cofactor matrix $\tilde{A}$ is defined as
Specifically, we see that
and in the form of a matrix,
Example:

Let $A$ be a 3x3 matrix given as
Each element of $\tilde{A}$ is
and in the form of a matrix,
Calculator
Enter a $3 \times 3$ matrix and press "Execute" button. Its cofactor matrix is displayed.

Matrix $A$
 1 2 3 1 2 3
Cofactor matrix $\tilde{A}$
 1 2 3 1 2 3

Example: $4 \times 4$ matrix
Find the cofactor matrix of

Let $M_{ij}$ be a submatrix given by removing $i$-th row and $j$-th column from $A$,
The determinant of $M_{ij}$ is respectively
(See 3x3 determinant). Each element of the cofactor matrix $\tilde{A}$ is defined as $$\tilde{a}_{ij} = (-1)^{i+j}|M_{ji}|$$ Specifically, we see that
Calculator
Enter a $4 \times 4$ matrix and press "Execute" button. Its cofactor matrix is displayed.

Matrix $A$
 1 2 3 4 1 2 3 4

Cofactor matrix $\tilde{A}$
 1 2 3 4 1 2 3 4

Definition of cofactor matrix
Let $a_{ij}$ be the $i$-th row and $j$-th column element of an $n \times n$ matrix $A$, and $M_{ij}$ be the sub-matrix obtained by removing the $i$-th row and $j$-column:
The $i$-th row and $j$-th column element of the cofactor matrix is defined as
, where $|M_{ji}|$ is the determinant of the sub-matrix $M_{ji}$. Representing the cofactor matrix in matrix form, we have