Question

Find the inverse of the given matrix
$\left[\begin{array}{cc}2& -2\\ 4& 3\end{array}\right]$

Answer 1

Let $A=\left[\begin{array}{cc}2& -2\\ 4& 3\end{array}\right]$, we have, $|A|\left[\begin{array}{cc}2& -2\\ 4& 3\end{array}\right]=6-(-8)=14$
Cofactors of A are $A_{11}=3,=-4,A_{21}=2,A_{22}=2$
$\therefore adj\left(A\right)={\left[\begin{array}{cc}3& -4\\ 2& 2\end{array}\right]}^T=\left[\begin{array}{cc}3& 2\\ -4& 2\end{array}\right]$
Now, $A^{-1}=\frac{1}{|A|}\left(adjA\right)=\frac{1}{14}\left[\begin{array}{cc}3& 2\\ -4& 2\end{array}\right]=\left[\begin{array}{cc}\frac{3}{14}& \frac{2}{14}\\ -\frac{4}{14}& \frac{2}{14}\end{array}\right]=\left[\begin{array}{cc}\frac{3}{14}& \frac{1}{7}\\ -\frac{2}{7}& \frac{1}{7}\end{array}\right]$
Note If determinant of any matrix is zero, then its inverse does not exist.