Let $A=\left[{a}_{ij}\right]$ be a square matrix of order $n$ . The adjoint of a matrix $A$ is the transpose of the cofactor matrix of $A$ . It is denoted by adj $A$ . An adjoint matrix is also called an adjugate matrix.

Example:

Find the adjoint of the matrix.

$A=\left[\begin{array}{ccc}3& \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1& -1\\ 2& -2& \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\\ 1& \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}2& -1\end{array}\right]$

To find the adjoint of a matrix, first find the cofactor matrix of the given matrix. Then find the transpose of the cofactor matrix.

Cofactor of $\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}3=$ ${A}_{11}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}|\begin{array}{rr}-2& 0\\ 2& -1\end{array}|=2$

Cofactor of $\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1=$ ${A}_{12}=-|\begin{array}{rr}2& 0\\ 1& -1\end{array}|=2$

Cofactor of $-1=$ ${A}_{13}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}|\begin{array}{rr}2& -2\\ 1& 2\end{array}|=6$

Cofactor of $\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}2=$ ${A}_{21}=-|\begin{array}{rr}1& -1\\ 2& -1\end{array}|=-1$

Cofactor of $-2=$ ${A}_{22}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}|\begin{array}{rr}3& -1\\ 1& -1\end{array}|=-2$

Cofactor of $\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0=$ ${A}_{23}=-|\begin{array}{rr}3& 1\\ 1& 2\end{array}|=-5$

Cofactor of $\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1=$ ${A}_{31}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}|\begin{array}{rr}1& -1\\ -2& 0\end{array}|=-2$

Cofactor of $\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}2=$ ${A}_{32}=-|\begin{array}{rr}3& -1\\ 2& 0\end{array}|=-2$

Cofactor of $-1=$ ${A}_{33}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}|\begin{array}{rr}3& 1\\ 2& -2\end{array}|=-8$

The cofactor matrix of $A$ is $\left[{A}_{ij}\right]=\left[\begin{array}{rrr}2& 2& 6\\ -1& -2& -5\\ -2& -2& -8\end{array}\right]$

Now find the transpose of ${A}_{ij}$ .

$\begin{array}{l}adj\text{\hspace{0.17em}}A={\left({A}_{ij}\right)}^{T}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\left[\begin{array}{rrr}2& -1& -2\\ 2& -2& -2\\ 6& -5& -8\end{array}\right]\end{array}$