Given A = $\left[\begin{array}{cc}2& -3\\ -4& 7\end{array}\right]$
|A| = 2 × - [(- 4) × (- 3)] = 2 ⇒ ${\mathit{A}}^{-1}$ exist.
Cofactors of matrix A are
${\mathit{C}}_{11}$ = 7 , ${\mathit{C}}_{12}$ = - 4 , ${\mathit{C}}_{21}$ = - 3 , ${\mathit{C}}_{22}$ = 2
Minors of matrix A are
${\mathit{M}}_{11}$ = ${\left(-1\right)}^{1+1}$ × 7 = 7
${\mathit{M}}_{12}$ = ${\left(-1\right)}^{1+2}$ × (- 4) = 4
${\mathit{M}}_{21}$ = ${\left(-1\right)}^{3+1}$ × (- 3) = 3
${\mathit{M}}_{22}$ = ${\left(-1\right)}^{2+2}$ × 2 = 2
${\mathit{A}}^{-1}$ = $\frac{1}{2}\left(\begin{array}{cc}7& 3\\ 4& 2\end{array}\right)$
2 ${\mathit{A}}^{-1}$ = $\left(\begin{array}{cc}7& 3\\ 4& 2\end{array}\right)$ ... (i)
9I - A = 9 $\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)-\left(\begin{array}{cc}2& -3\\ -4& 7\end{array}\right)$
= $\left(\begin{array}{cc}7& 3\\ 4& 2\end{array}\right)$ ... (ii)
⇒ $2{\mathit{X}}^{-1}$ = 9I - A from (i) and (ii)