Chapter 7, Problem 12RE

Solution

To calculate: The inverse of the matrix if it has one and use the multiplication to suppose the answer.

The invers is $\frac{1}{5}\cdot \left(\begin{array}{ccc}-2& 1& 1\\ -1& -2& 3\\ 3& 1& 1\end{array}\right)$ .

Given Information:

The matrix is defined as,

  $A=\left[\begin{array}{ccc}-1& 0& 1\\ 2& -1& 1\\ 1& 1& 1\end{array}\right]$

Calculation:

Consider the given metrics,

  $A=\left[\begin{array}{ccc}-1& 0& 1\\ 2& -1& 1\\ 1& 1& 1\end{array}\right]$

Find the determinant.

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

Substitute the values in the inverse formula.

  $\begin{array}{c}{A}^{-1}=\frac{1}{\left|A\right|}\text{adj}\left[A\right]\\ =\frac{1}{5}\left[\begin{array}{ccc}-2& 1& 1\\ -1& -2& 3\\ 3& 1& 1\end{array}\right]\end{array}$

Now, multiply the obtained inverse with the given to cross check the answer and use the calculator to find the multiplication.

  $\begin{array}{c}A{A}^{-1}=\frac{1}{5}\cdot \left[\begin{array}{ccc}-1& 0& 1\\ 2& -1& 1\\ 1& 1& 1\end{array}\right]\cdot \left[\begin{array}{ccc}-2& 1& 1\\ -1& -2& 3\\ 3& 1& 1\end{array}\right]\\ =\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\end{array}$

Thus, the inverse is exists.

Hence, the inverse of the given metrics is $\frac{1}{5}\cdot \left(\begin{array}{ccc}-2& 1& 1\\ -1& -2& 3\\ 3& 1& 1\end{array}\right)$ .