Chapter 11.4, Problem 65AYU

To determine

To show: The matrix has no inverse.

Answer to Problem 65AYU

Solution:

A is a singular matrix and has no inverse.

Explanation of Solution

Given:

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

Calculation:

Step 1 : Construction of $\left[A\mid I_n\right]$

  

Step 2 : Sequentially transforming the matrix $\left[A\mid I_n\right]$ into reduced row echelon form.

  

By definition, if we interchange the rows of a matrix, then the value of the matrix remains unchanged (i.e.) interchanging Rows 1 and 3.

  

The matrix $\left[A\mid I_n\right]$ is sufficiently reduced for it to be clear that the identity matrix cannot appear to the left of the vertical bar. So A is a singular matrix and has no inverse.