Chapter 8, Problem 8PS

To determine

Find x such that given matrix is equal to its own inverse

Answer to Problem 8PS

The value of x for which above equation $A=A^{-1}$ is valid is 4.

Explanation of Solution

Given:

The matrix,

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

Consider the matrix,

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

Find the value of x for which $A=A^{–1}$ . Pre-multiply both sides of the equation by A,

  $\begin{array}{l}A\left(A\right)=A{A}^{-1}\hfill \\ A•A=I\hfill \\ \left[\begin{array}{cc}3& x\\ -2& -3\end{array}\right]\left[\begin{array}{cc}3& x\\ -2& -3\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\hfill \\ \left[\begin{array}{cc}9-2x& 0\\ 0& 9-2x\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\hfill \end{array}$

Equate the elements as done below,

  $\begin{array}{l}\begin{array}{l}9–2x=1\hfill \\ 2x=9–1\hfill \\ 2x=8\hfill \end{array} \\ x=4\end{array}$

Thus, the value of x for which above equation $A=A^{-1}$ is valid is 4.