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

Question:

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

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

Solution:

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

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

By expanding along $C_{1}$, we have:

$|A|=1(8-6)-0+3(3-4)=2-3=-1$

Now,

$A_{11}=8-6=2, A_{12}=-(0+9)=-9, A_{13}=0-6=-6$

$A_{21}=-(-4+4)=0, A_{22}=4-6=-2, A_{23}=-(-2+3)=-1$

$A_{31}=3-4=-1, A_{32}=-(-3-0)=3, A_{33}=2-0=2$

$\therefore \operatorname{adj} A=\left[\begin{array}{lll}2 & 0 & -1 \\ -9 & -2 & 3 \\ -6 & -1 & 2\end{array}\right]$

$\therefore A^{-1}=\frac{1}{|A|}$ adj $A=-\left[\begin{array}{lll}2 & 0 & -1 \\ -9 & -2 & 3 \\ -6 & -1 & 2\end{array}\right]=\left[\begin{array}{lll}-2 & 0 & 1 \\ 9 & 2 & -3 \\ 6 & 1 & -2\end{array}\right]$