Solve inverse matrix extensively

For a $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the determinant is $ad - bc$.

Since $\det(A) = 1 \neq 0$, the inverse of the matrix exists. If the determinant were 0, the inverse would not exist.

For a $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the adjugate matrix is found by swapping the elements on the main diagonal ($a$ and $d$) and changing the signs of the off-diagonal elements ($b$ and $c$).

The formula for the inverse of a matrix is $A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$.

The inverse of the matrix $A = \begin{bmatrix} 3 & 1 \\ 5 & 2 \end{bmatrix}$ is: