Using elementary transformations, find the inverse of the matrix`[[3, 1], [5, 2]]`

Explanation

To find the inverse of the matrix \(\begin{bmatrix} 3 & 1 \\ 5 & 2 \end{bmatrix}\) using elementary transformations, we will follow these steps:

### Step 1: Set up the augmented matrix
We start with the matrix \(A = \begin{bmatrix} 3 & 1 \\ 5 & 2 \end{bmatrix}\) and augment it with the identity matrix \(I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\). This gives us the augmented matrix:

\[ \begin{bmatrix} 3 & 1 & | & 1 & 0 \\ 5 & 2 & | & 0 & 1 \end{bmatrix} \]
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