Answer
The matrix $A$ has no inverse.
Work Step by Step
In order to calculate the inverse of an $n$ by $n$ non-singular matrix $A$, we will proceed with the following steps:
Step -1. Transform the given matrix $A$ into this form $\left[A|I_{n}\right]$ as follows:
$\left[A|I_{n}\right]$ = $\left[\begin{array}{ll|ll}
{-3}&{\dfrac{1}{2}}&{1}&{0}\\
{6}&{-1}&{0}&{1}\end{array}\right]$
Step-2. Transform the matrix $\left[A|I_{n}\right]$ into reduced row-echelon form by using the row operations: $R_{2}=2r_{1}+r_{2}$
$\left[\begin{array}{rr|rr}{-3}&{\dfrac{1}{2}}&{1}&{0}\\
{0}&{0}&{2}&{1}\end{array}\right]$
We see that the zeros on the left of the vertical line in Row-2 make it impossible to obtain a leading nonzero entry in Column-2. This means that the identity matrix does not occur on the left of the reduced row-echelon form of $\left[A|I_{n}\right]$, so the matrix $A$ has no inverse.
