Chapter 7.2, Problem 37E

Solution

To calculate: The inverse of the matrix if it exists, or explain that the inverse does not exists.

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

The obtained result of the inverse matrix does not exists.

Given Information:

The matrix is defined as,

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

Formula Used:

Suppose that $A$ is general matrix of order 3 and defined as,

  $A=\left[\begin{array}{ccc}{A}_{11}& A_{12}& A_{13}\\ A_{21}& A_{22}& A_{22}\\ A_{31}& A_{32}& A_{33}\end{array}\right]$

The inverse of the matrix is defined as,

  $A^{-1}=\frac{1}{\left|A\right|}\text{adj}\left(A\right)$

Where, $\left|A\right|$ is representing the determinant and $adj\left(A\right)$ is defined as,

  $\begin{array}{c}adjA={\left[\begin{array}{ccc}{A}_{11}& A_{12}& A_{13}\\ A_{21}& A_{22}& A_{23}\\ A_{31}& A_{32}& A_{33}\end{array}\right]}^T\\ =\left[\begin{array}{ccc}{A}_{11}& A_{21}& A_{31}\\ A_{12}& A_{22}& A_{32}\\ A_{13}& A_{23}& A_{33}\end{array}\right]\end{array}$

And, $\left[\begin{array}{ccc}{A}_{11}& A_{12}& A_{13}\\ A_{21}& A_{22}& A_{23}\\ A_{31}& A_{32}& A_{33}\end{array}\right]$ is a cofactor matrix of $A$ .

Calculation:

Consider the given matrix,

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

First, find the determinant.

  $\begin{array}{c}\left|A\right|=\left|\begin{array}{ccc}1& 2& -1\\ 2& -1& 3\\ 3& 1& 2\end{array}\right|\\ =1\left[-2-3\right]-2\left[4-9\right]-1\left[2+3\right]\\ =-5+10-5\\ =0\end{array}$

As the obtained determinant of the matrix is 0, which is making the inverse of the matrix undefined. So, the inverse does not exists.

Hence, the inverse of the matrix does not exists.