Question

Find the adjoint of matrix $A=\left[\begin{array}{ccc}2& 0& 1\\ 3& 1& 2\\ -1& 1& 2\end{array}\right]$

Answer 1

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

Adjoint of a matrix is difined as the transpose of the co-factor matrix.

If $C$ be the co-factor matrix then,

$C=\left[\begin{array}{ccc}{C}_{11}& C_{12}& C_{13}\\ C_{21}& C_{22}& C_{23}\\ C_{31}& C_{32}& C_{33}\end{array}\right]$

$C_{11}=\left|\begin{array}{cc}1& 2\\ 1& 2\end{array}\right|=0$ $C_{12}=-\left|\begin{array}{cc}3& 2\\ -1& 2\end{array}\right|=-8$

$C_{13}=\left|\begin{array}{cc}3& 1\\ -1& 1\end{array}\right|=4$ $C_{21}=-\left|\begin{array}{cc}0& 1\\ 1& 2\end{array}\right|=1$

$C_{22}=\left|\begin{array}{cc}2& 1\\ -1& 2\end{array}\right|=5$ $C_{23}=-\left|\begin{array}{cc}2& 0\\ -1& 1\end{array}\right|=-2$

$C_{31}=\left|\begin{array}{cc}0& 1\\ 1& 2\end{array}\right|=-1$ $C_{32}=-\left|\begin{array}{cc}2& 1\\ 3& 2\end{array}\right|=-1$

$C_{33}=\left|\begin{array}{cc}2& 0\\ 3& 1\end{array}\right|=2$

Now, $C=\left[\begin{array}{ccc}0& -8& 4\\ 1& 5& -2\\ -1& -1& 2\end{array}\right]$

$C^T=\left[\begin{array}{ccc}0& 1& -1\\ -8& 5& -1\\ 4& -2& 2\end{array}\right]$

$\therefore adjA=\left[\begin{array}{ccc}0& 1& -1\\ -8& 5& -1\\ 4& -2& 2\end{array}\right]$