Chapter 7, Problem 11RE

Solution

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

The invers is $\left(\begin{array}{cccc}-2& -5& 6& -1\\ 0& -1& 1& 0\\ 10& 24& -27& 4\\ -3& -7& 8& -1\end{array}\right)$ .

Given Information:

The matrix is defined as,

  $A=\left[\begin{array}{cccc}1& 2& 0& -1\\ 2& -1& 1& 2\\ 2& 0& 1& 2\\ -1& 1& 1& 4\end{array}\right]$

Calculation:

Consider the given metrics,

  $A=\left[\begin{array}{cccc}1& 2& 0& -1\\ 2& -1& 1& 2\\ 2& 0& 1& 2\\ -1& 1& 1& 4\end{array}\right]$

Find the determinant.

  $\begin{array}{c}\left|\begin{array}{cccc}1& 2& 0& -1\\ 2& -1& 1& 2\\ 2& 0& 1& 2\\ -1& 1& 1& 4\end{array}\right|=1\left|\begin{array}{ccc}-1& 1& 2\\ 0& 1& 2\\ 1& 1& 4\end{array}\right|-2\left|\begin{array}{ccc}2& 1& 2\\ 2& 1& 2\\ -1& 1& 4\end{array}\right|+0\left|\begin{array}{ccc}2& -1& 2\\ 2& 0& 2\\ -1& 1& 4\end{array}\right|+1\left|\begin{array}{ccc}2& -1& 1\\ 2& 0& 1\\ -1& 1& 1\end{array}\right|\\ =-2+0+0+3\\ =1\end{array}$

Substitute the values in the inverse formula.

  $\begin{array}{c}{A}^{-1}=\frac{1}{\left|A\right|}\text{adj}\left[A\right]\\ =\frac{1}{1}\left[\begin{array}{cccc}-2& -5& 6& -1\\ 0& -1& 1& 0\\ 10& 24& -27& 4\\ -3& -7& 8& -1\end{array}\right]\end{array}$

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

  $\begin{array}{c}A{A}^{-1}=\left(\begin{array}{cccc}1& 2& 0& -1\\ 2& -1& 1& 2\\ 2& 0& 1& 2\\ -1& 1& 1& 4\end{array}\right)\cdot \left(\begin{array}{cccc}-2& -5& 6& -1\\ 0& -1& 1& 0\\ 10& 24& -27& 4\\ -3& -7& 8& -1\end{array}\right)\\ =\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\end{array}$

Thus, the inverse is exists.

Hence, the inverse of the given metrics is $\left(\begin{array}{cccc}-2& -5& 6& -1\\ 0& -1& 1& 0\\ 10& 24& -27& 4\\ -3& -7& 8& -1\end{array}\right)$ .