Q. 61 - Precalculus Enhanced with Graphing Utilities

Question

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

$\left[\begin{array}{ll}4& 2\\ 2& 1\end{array}\right]$

Step-by-Step Solution

Answer

The given matrix has no inverse.

Step 1. Given information

A matrix is given as,

$\left[\begin{array}{ll}4& 2\\ 2& 1\end{array}\right]$

Step 2. Finding the inverse.

To find the inverse,

$$\begin{gathered} \left[A\mid I_2\right]=\left[\begin{array}{llll}4& 2& 1& 0\\ 2& 1& 0& 1\end{array}\right] \\ {R}_1=\frac{1}{4}{r}_1, R_2=r_2-2r_1 \\ \left[\begin{array}{llll}4& 2& 1& 0\\ 2& 1& 0& 1\end{array}\right]\to \left[\begin{array}{cccc}1& \frac{1}{2}& \frac{1}{4}& 0\\ 2& 1& 0& 1\end{array}\right]\to \left[\begin{array}{cccc}1& \frac{1}{2}& \frac{1}{4}& 0\\ 0& 0& -\frac{1}{2}& 1\end{array}\right] \end{gathered}$$

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