Using elementary transformations, find the inverse of the matrix`[[2,-3],[-1, 2]]`

Explanation

To find the inverse of the matrix \( A = \begin{pmatrix} 2 & -3 \\ -1 & 2 \end{pmatrix} \) using elementary transformations, we will augment the matrix \( A \) with the identity matrix \( I \) of the same order. The identity matrix for a \( 2 \times 2 \) matrix is \( I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \).

### Step 1: Set up the augmented matrix
We start with the augmented matrix \( [A | I] \):
\[
\begin{pmatrix}
2 & -3 & | & 1 & 0 \\
-1 & 2 & | & 0 & 1
...