Chapter 7.2, Problem 35E

Solution

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

  $\left[\begin{array}{cc}2& 3\\ 2& 2\end{array}\right]$

  $\left[\begin{array}{cc}-1& 1.5\\ 1& -1\end{array}\right]$

Given Information:

The matrix is defined as,

  $\left[\begin{array}{cc}2& 3\\ 2& 2\end{array}\right]$

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,

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

First, find the determinant.

  $\begin{array}{c}\left|A\right|=\left|\begin{array}{cc}2& 3\\ 2& 2\end{array}\right|\\ =4-6\\ =-2\end{array}$

Now, find the ad-joint matrix.

  $adjA=\left[\begin{array}{cc}2& -3\\ -2& 2\end{array}\right]$

Put the above the value in the formula of inverse.

  $\begin{array}{c}{A}^{-1}=\frac{1}{-2}\left[\begin{array}{cc}2& -3\\ -2& 2\end{array}\right]\\ =\left[\begin{array}{cc}-1& 1.5\\ 1& -1\end{array}\right]\end{array}$

Hence, the inverse of the matrix is $\left[\begin{array}{cc}-1& 1.5\\ 1& -1\end{array}\right]$ .