There are mainly two ways to obtain the inverse matrix.
One is to use Gauss-Jordan elimination and the other is to use the adjugate matrix.
We employ the latter, here.

The inverse matrix has the property that
it is equal to the product of the reciprocal of the determinant and the adjugate matrix.

where $\tilde{A}$ is the adjugate matrix of $A$, and
$|A|$ is the determinant.
The determinant of $A$ can be obtained by using the cofactor expansion.
The cofactor expansion along the first column is