The inverse of the matrix $A=\left[\begin{array}{ll}2 & 3 \\ 1 & 2\end{array}\right]$ is:

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

$A=\left[\begin{array}{ll}2 & 3 \\ 1 & 2\end{array}\right]$

|A| = 2 × 2 - 3 × 1 = 4 - 3

|A| = 1

finding adjoint of A → Cofactors

$C_{11}=(-1)^{1+1} 2=2 \quad C_{12}=(-1)^{1+2} 1=-1$

$C_{21}=(-1)^{2+1} 3=-3 \quad C_{22}=(-1)^{2+2} 2=2$

Adj A = $\left[\begin{array}{ll}c_{11} & c_{12} \\ c_{21} & c_{22}\end{array}\right]^{T} = \left[\begin{array}{cc}2 & -1 \\ -3 & 2\end{array}\right]^{\top}=\left[\begin{array}{cc}2 & -3 \\ -1 & 2\end{array}\right]$

$A^{-1}=\frac{1}{|A|} Adj ~A$

$\frac{1}{1}\left[\begin{array}{cc}2 & -3 \\ -1 & 2\end{array}\right] = A^{-1}$