Q. 62 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 61– 66, show that each matrix has no inverse. 

$\left[\begin{array}{rr}-3& \frac{1}{2}\\ 6& -1\end{array}\right]$

Step-by-Step Solution

Answer

The given matrix has no inverse.

Step 1. Given information

A matrix is given as,

$\left[\begin{array}{rr}-3& \frac{1}{2}\\ 6& -1\end{array}\right]$

Step 2. Finding the inverse.

To find the inverse,

$$\begin{gathered} \left[A\mid I_2\right]=\left[\begin{array}{ccc}-3& \frac{1}{2}\mid \begin{array}{cc}1& 0\\ 6& -1\end{array}& 1\end{array}\right] \\ \left\{R_1\to \frac{R_1}{-3}\right\} \\ \left[\begin{array}{rrrr}1& -\frac{1}{6}& -\frac{1}{3}& 0\\ 6& -1& 0& 1\end{array}\right] \\ \left\{R_2\to R_2-6R_1\right\} \\ \left|\begin{array}{cccc}1& -\frac{1}{6}& -\frac{1}{3}& 0\\ 0& 0& 2& 1\end{array}\right| \end{gathered}$$

Since the left side does not represent an identity matrix, therefore, inverse does not exist.