Question

Find the inverse of the given matrix
$\left[\begin{array}{ccc}1& 2& 3\\ 0& 2& 4\\ 0& 0& 5\end{array}\right]$

Answer 1

Let $A=\left[\begin{array}{ccc}1& 2& 3\\ 0& 2& 4\\ 0& 0& 5\end{array}\right]$
We have, $|A|=1(10-0)-2(0-0)+3(0-0)=10$
Now, cofactors of A are
$A_{11}=10-0=10,A_{12}=-(0-0)=0,A_{13}=0-0=0A_{21}=(10-0)=-10,A_{22}=5-0=5,A_{23}=-(0-0)=0,A_{31}=8-6=2,A_{32}=-(4-0)=-4,A_3=2-0=2,$
$\therefore adj\left(A\right)={\left[\begin{array}{ccc}10& 0& 0\\ -10& 5& 0\\ 2& -4& 2\end{array}\right]}^T=\left[\begin{array}{ccc}10& -10& 2\\ 0& 5& -4\\ 0& 0& 2\end{array}\right]$
Now, $A^{-1}=\frac{1}{|A|}=\left(adjA\right)=\frac{1}{10}\left[\begin{array}{ccc}10& -10& 2\\ 0& 5& -4\\ 0& 0& 2\end{array}\right]=\left[\begin{array}{ccc}1& -1& \frac{1}{5}\\ 0& \frac{1}{2}& -\frac{2}{5}\\ 0& 0& \frac{1}{5}\end{array}\right]$