Find the inverse of each of the matrices (if it exists)

Question:

Find the inverse of each of the matrices (if it exists)

$\left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 2 & 4 \\ 0 & 0 & 5\end{array}\right]$

Solution:

Let $A=\left[\begin{array}{lll}1 & 2 & 3 \\ 0 & 2 & 4 \\ 0 & 0 & 5\end{array}\right]$.

We have.

$|A|=1(10-0)-2(0-0)+3(0-0)=10$

Now,

$A_{11}=10-0=10, A_{12}=-(0-0)=0, A_{13}=0-0=0$

$A_{21}=-(10-0)=-10, A_{22}=5-0=5, A_{23}=-(0-0)=0$

$A_{31}=8-6=2, A_{32}=-(4-0)=-4, A_{33}=2-0=2$

$\therefore a d j A=\left[\begin{array}{ccc}10 & -10 & 2 \\ 0 & 5 & -4 \\ 0 & 0 & 2\end{array}\right]$

$\therefore A^{-1}=\frac{1}{|A|}$ adjA $=\frac{1}{10}\left[\begin{array}{ccc}10 & -10 & 2 \\ 0 & 5 & -4 \\ 0 & 0 & 2\end{array}\right]$