Question

Find the adjoint matrix of $=\left(\begin{array}{ccc}2& 5& 3\\ 3& 1& 2\\ 1& 2& 1\end{array}\right)$
$\left[A_{ij}\right]=\left[\begin{array}{ccc}-3& -1& 5\\ 1& -1& 1\\ 7& 5& -13\end{array}\right]adjA={\left[A_{ij}\right]}^T$

Answer 1

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

$C_{11}={(-1)}^{1+1}{M}_{11}=\left[\begin{array}{cc}1& 2\\ 2& 1\end{array}\right]=1-4=-3$

$C_{12}={(-1)}^{1+2}{M}_{12}=-\left[\begin{array}{cc}3& 2\\ 1& 1\end{array}\right]=-(3-2)=-1$

$C_{13}={(-1)}^{1+3}{M}_{13}=\left[\begin{array}{cc}3& 1\\ 1& 2\end{array}\right]=6-1=5$

$C_{21}={(-1)}^{2+1}{M}_{21}=-\left[\begin{array}{cc}5& 3\\ 2& 1\end{array}\right]=-(5-6)=1$

$C_{22}={(-1)}^{2+2}{M}_{22}=\left[\begin{array}{cc}2& 3\\ 1& 1\end{array}\right]=2-3=-1$

$C_{23}={(-1)}^{2+3}{M}_{23}=-\left[\begin{array}{cc}2& 5\\ 1& 2\end{array}\right]=-(4-5)=1$

$C_{31}={(-1)}^{3+1}{M}_{31}=\left[\begin{array}{cc}5& 3\\ 1& 2\end{array}\right]=10-3=7$

$C_{32}={(-1)}^{3+2}{M}_{32}=-\left[\begin{array}{cc}2& 3\\ 3& 2\end{array}\right]=-(4-9)=5$

$C_{33}={(-1)}^{3+3}{M}_{33}=\left[\begin{array}{cc}2& 5\\ 3& 1\end{array}\right]=2-15=-13$

$C_{ij}=\left[\begin{array}{ccc}-3& -1& 5\\ 1& -1& 1\\ 7& 5& -13\end{array}\right]$

Adj$\left(A\right)={C_{ij}}^T={\left[\begin{array}{ccc}-3& -1& 5\\ 1& -1& 1\\ 7& 5& -13\end{array}\right]}^T$

$=\left[\begin{array}{ccc}-3& 1& 7\\ -1& -1& 5\\ 5& 1& -13\end{array}\right]$