## Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

The matrix $A$ has no inverse.
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.