# Adjoint of a Square Matrix

### Adjoint of a Square Matrix:

The adjoint of a square matrix $A$, denoted as $\text{adj}\left(A\right)$ or ${A}^{\ast }$, is a matrix formed by taking the transpose of the matrix of cofactors (also known as the cofactor matrix) of $A$.

### Calculation:

For a square matrix $A$ of size $n×n$:

1. Cofactor Matrix: Calculate the matrix of cofactors $C$ for matrix $A$. Each element ${c}_{ij}$ of $C$ is the determinant of the submatrix obtained by removing the $i$th row and $j$th column of $A$, multiplied by $\left(-1{\right)}^{i+j}$.
2. Transpose: Transpose the matrix $C$ to obtain the adjoint $\text{adj}\left(A\right)$ or ${A}^{\ast }$.

### Example:

Consider a matrix $A$:

$A=\left[\begin{array}{cc}2& 3\\ 5& 4\end{array}\right]$
1. Cofactor Matrix:
$C=\left[\begin{array}{cc}4& -5\\ -3& 2\end{array}\right]$
1. Transpose:
$\text{adj}\left(A\right)={C}^{T}=\left[\begin{array}{cc}4& -3\\ -5& 2\end{array}\right]$

1. Inverses: For a non-singular matrix $A$, ${A}^{-1}=\frac{1}{\text{det}\left(A\right)}\text{adj}\left(A\right)$, where $\text{det}\left(A\right)$ is the determinant of $A$.
2. Non-Singular Matrix: If $\text{det}\left(A\right)\mathrm{\ne }0$, $A$ is non-singular, and its adjoint exists.
3. Product with Matrix: For any matrix $B$, $A\left(\text{adj}\left(A\right)B\right)=\left(\text{adj}\left(A\right)BA\right)=\left(\text{det}\left(A\right)I\right)B=\text{det}\left(A\right)B$