Question

Find the adjoint of given matrix.

$\left[\begin{array}{ccc}1& -1& 2\\ 2& 3& 5\\ -2& 0& 1\end{array}\right]$

Answer 1

Let $A=\left[\begin{array}{ccc}1& -1& 2\\ 2& 3& 5\\ -2& 0& 1\end{array}\right]$
Cofactors of elements of first row are $A_{11}=\left|\begin{array}{cc}3& 5\\ 0& 1\end{array}\right|=3-0=3,$
$A_{12}=-\left|\begin{array}{cc}2& 5\\ -1& 1\end{array}\right|=-(2+10)=-\left(12\right),A_{13}=\left|\begin{array}{cc}2& 3\\ -2& 0\end{array}\right|=0-(-6)=6$
Cofactors of elements of second row are
$A_{21}=-\left|\begin{array}{cc}-1& 2\\ 0& 1\end{array}\right|=-(-1-0)=1,A_{22}=\left|\begin{array}{cc}1& 2\\ -2& 1\end{array}\right|=(1+4)=5A_{23}=-\left|\begin{array}{cc}1& -1\\ -2& 0\end{array}\right|=-(0-2)=2$

Cofactors of elements of third row are
$A_{31}=\left|\begin{array}{cc}-1& 2\\ 3& 5\end{array}\right|=(-5-6)=-11,A_{32}=-\left|\begin{array}{cc}1& 2\\ 2& 5\end{array}\right|=-(5-4)=-1$
$A_{33}=\left|\begin{array}{cc}1& -1\\ 2& 3\end{array}\right|=3+2=5$
Hence, $a_{dj}\left(A\right)={\left[\begin{array}{ccc}3& -12& 6\\ 1& 5& -2\\ -11& -1& 5\end{array}\right]}^T=\left[\begin{array}{ccc}3& 1& -11\\ -12& 5& -1\\ 6& 2& 5\end{array}\right]$