Question

Find the inverse of the given matrix
$\left[\begin{array}{ccc}1& 0& 0\\ 3& 3& 0\\ 5& 2& -1\end{array}\right]$

Answer 1

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

We have, $|A|=\left[\begin{array}{ccc}1& 0& 0\\ 3& 3& 0\\ 5& 2& -1\end{array}\right]=1(-3-0)-0+0=-3$
Cofactors of A are
$A_{11}=-3-0=-3,A_{12}=-(-3-0)-3,A_{13}=6-15-9A_{21}=-(0-0)=0,A_{22}=-1-0=-1,A_{23}=-(2-0)=-2A_{31}=0-0=0,A_{32}=-(0-0)=0,A_{33}=3-0=3$
$\therefore adj\left(A\right)={\left[\begin{array}{ccc}-3& 3& -9\\ 0& -1& -2\\ 0& 0& 3\end{array}\right]}^T=\left[\begin{array}{ccc}-3& 0& 0\\ 3& -1& 0\\ -9& -2& 3\end{array}\right]$

Now, $A^{-1}=\frac{1}{|A|}\left(adjA\right)=\frac{1}{-3}\left[\begin{array}{ccc}-3& 0& 0\\ 3& -1& 0\\ -9& -2& 3\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ -1& \frac{1}{3}& 0\\ 3& \frac{2}{3}& -1\end{array}\right]$