Chapter 6.3, Problem 7E

In Exercises 9-18, determine whether B is the inverse of A by computing AB and BA.

$9.A=\left[\begin{array}{cc}1& 2\\ 1& 3\end{array}\right],B=\left[\begin{array}{cc}3& -2\\ -1& 1\end{array}\right]$

$10.A=\left[\begin{array}{cc}3& 2\\ 4& 3\end{array}\right],B=\left[\begin{array}{cc}3& -2\\ -4& 3\end{array}\right]$

$11.A=\left[\begin{array}{cc}3& 2\\ 1& 4\end{array}\right],B=\left[\begin{array}{cc}\frac{2}{5}& -\frac{1}{5}\\ -\frac{1}{10}& \frac{3}{10}\end{array}\right]$

$12.A=\left[\begin{array}{cc}2& -3\\ 4& -3\end{array}\right],B=\left[\begin{array}{cc}-\frac{1}{2}& \frac{1}{2}\\ -\frac{2}{3}& \frac{1}{3}\end{array}\right]$

$13.A=\left[\begin{array}{ccc}1& 0& -1\\ -1& 1& 1\end{array}\right],B=\left[\begin{array}{cc}2& 1\\ 1& 1\\ 1& 1\end{array}\right]$

$14.A=\left[\begin{array}{ccc}-2& 1& 3\\ 0& -1& 1\\ 1& 2& 0\end{array}\right],B=\frac{1}{8}\left[\begin{array}{ccc}-2& 6& 4\\ 1& -3& 2\\ 1& 5& 2\end{array}\right]$

$15.A=\left[\begin{array}{ccc}1& -2& 1\\ -8& 6& -2\\ 5& -3& 1\end{array}\right],B=\frac{1}{2}\left[\begin{array}{ccc}0& 1& 2\\ 1& 2& 3\\ 3& 1& 1\end{array}\right]$

$16.A=\left[\begin{array}{ccc}2& 3& 1\\ 1& 2& 3\\ 3& 2& 1\end{array}\right],B=\frac{1}{18}\left[\begin{array}{ccc}1& -5& 7\\ 7& 1& -5\\ -5& 7& 1\end{array}\right]$

$17.A=\left[\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ -1& 1& -1\end{array}\right],B=\frac{1}{4}\left[\begin{array}{ccc}5& -2& -1\\ 2& 0& 2\\ -3& 2& -1\end{array}\right]$

$18.A=\left[\begin{array}{ccc}1& -1& 0\\ 0& 1& -1\\ 1& 0& 1\end{array}\right],B=\frac{1}{2}\left[\begin{array}{ccc}1& 1& 1\\ -1& 1& 1\\ -1& -1& 1\end{array}\right]$