If $A=\left[\begin{array}{rr}2 & 3 \\ 5 & -2\end{array}\right]$ then :

$A^{-1}=\frac{1}{19} A$

$A=\left[\begin{array}{cc}2 & 3 \\ 5 & -2\end{array}\right]$

finding cofactors of A 

C₁₁ = -2

C₁₂ = -5

C₂₁ = -3

C₂₂ = 2

so  Adj A = $\left[\begin{array}{ll}C_{11} & C_{12} \\ C_{21} & C_{22}\end{array}\right]^{T}=\left[\begin{array}{cc}-2 & -5 \\ -3 & 2\end{array}\right]^{T}=\left[\begin{array}{cc}-2 & -3 \\ -5 & 2\end{array}\right]$ = Adj A

|A| = 2(-2) - 5(3)

= -4 - 15

= -19

so $A^{-1}=\frac{1}{|A|} adj~A=\frac{-1}{19}\left[\begin{array}{ll}-2 & -3 \\ -5 & 2\end{array}\right]$

$\Rightarrow \frac{1}{19}\left[\begin{array}{rr}2 & 3\\ 5 & -2\end{array}\right]$