The given matrix A is a 2x2 matrix. To find its inverse, we use the formula for the inverse of a 2x2 matrix $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$, which is $A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$. The determinant (ad - bc) must not be zero.

For matrix A = $\left[\begin{array}{cc}2& 3\\ 1& 2\end{array}\right]$, the determinant is (2)(2) - (3)(1) = 4 - 3 = 1. Therefore, the inverse is $\frac{1}{1}\left[\begin{array}{cc}2& -3\\ -1& 2\end{array}\right]=\left[\begin{array}{cc}2& -3\\ -1& 2\end{array}\right]$.

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