Q. 66 - Precalculus Enhanced with Graphing Utilities

Question

 Show that each matrix has no inverse. 

$\left[\begin{array}{ccc}1& 1& -3\\ 2& -4& 1\\ -5& 7& 1\end{array}\right]$

Step-by-Step Solution

Answer

The given matrix inverse is not possible

Step 1.Given information

Given matrix $\left[\begin{array}{ccc}1& 1& -3\\ 2& -4& 1\\ -5& 7& 1\end{array}\right]$

Step 2.Inverse matrix

Consider the matrix .To check the inverse of the matrix we have to reduce to row echelon .

$$\begin{gathered} \left[A\right|I_3]=\left[\begin{array}{ccc}1& 1& -3\\ 2& -4& 1\\ -5& 7& 1\end{array}\left|\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right|\right] \\ \{R_2\to R_2-2R_1;R_3\to R_3+5R_1 \\ = \left[\begin{array}{ccc}1& 1& -3\\ 0& -6& 7\\ 0& 12& -14\end{array}\left|\begin{array}{ccc}1& 0& 0\\ -2& 1& 0\\ 5& 0& 1\end{array}\right|\right] \\ \{R_3\to R_3+2R_2\} \\ =\left[\begin{array}{ccc}1& 1& -3\\ 0& -6& 7\\ 0& 0& 0\end{array}\left|\begin{array}{ccc}1& 0& 0\\ -2& 1& 0\\ 1& 2& 1\end{array}\right|\right] \end{gathered}$$

step 3.The solution

The matrix $\left[A|I_3\right]$  is reduced the identical matrix cannot appeared .

So inverse of the given matrix is not possible.