Answer
The identity matrix can not be obtained on the left of the reduced row echelon form of $\left[A|I_{n}\right]$.
Thus, A has no inverse.
Work Step by Step
Procedure for Finding the Inverse of a Matrix
STEP 1: Form the matrix $\left[A|I_{n}\right]$
STEP 2: Transform the matrix $\left[A|I_{n}\right]$ into reduced row echelon form.
STEP 3: The reduced row echelon form of $\left[A|I_{n}\right]$ will contain the identity matrix $I_{n}$ on the left of the vertical bar;
the $n$ by $n$ matrix on the right of the vertical bar is the inverse of $A. $
If the identity matrix can not be obtained on the left, A has no inverse.
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$\left[A|I_{n}\right]=\left[\begin{array}{rr|rr}
{-3}&{\displaystyle \frac{1}{2}}&{1}&{0}\\
{6}&{-1}&{0}&{1}\end{array}\right]\rightarrow( R_{2}=2r_{1}+r_{2})\rightarrow$
$\rightarrow\left[\begin{array}{rr|rr}
{-3}&{\displaystyle \frac{1}{2}}&{1}&{0}\\
{0}&{0}&{2}&{1}\end{array}\right]$
The zeros on the left of the bar in the second row
make it impossible to obtain a leading nonzero entry in the second column.
The identity matrix can not be obtained on the left of the reduced row echelon form of $\left[A|I_{n}\right]$.
Thus, A has no inverse.
