Show that AB = BA, where A = [-23-1-12-1-69-4],B=[13-122-130-1].

Exercise 2.3 | Q 3 | Page 55

QUESTION

Show that AB = BA, where A = ${\displaystyle \left[\begin{array}{ccc}-2& 3& -1\\ -1& 2& -1\\ -6& 9& -4\end{array}\right],\text{B}=\left[\begin{array}{ccc}1& 3& -1\\ 2& 2& -1\\ 3& 0& -1\end{array}\right]}$.

SOLUTION

AB = ${\displaystyle \left[\begin{array}{ccc}-2& 3& -1\\ -1& 2& -1\\ -6& 9& -4\end{array}\right]\left[\begin{array}{ccc}1& 3& -1\\ 2& 2& -1\\ 3& 0& -1\end{array}\right]}$

= ${\displaystyle \left[\begin{array}{ccc}-2+6-3& -6+6-0& 2-3+1\\ -1+4-3& -3+4-0& 1-2+1\\ -6+18-12& -18+18+0& 6-9+4\end{array}\right]}$

∴ AB = ${\displaystyle \left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]}$         ...(i)

BA = ${\displaystyle \left[\begin{array}{ccc}1& 3& -1\\ 2& 2& -1\\ 3& 0& -1\end{array}\right]\left[\begin{array}{ccc}-2& 3& -1\\ -1& 2& -1\\ -6& 9& -4\end{array}\right]}$

= ${\displaystyle \left[\begin{array}{ccc}2-3+6& 3+6-9& -1-3+4\\ -4-2+6& 6+4-9& -2-2+4\\ -6+0+6& 9+0-9& 3+0+4\end{array}\right]}$

∴ BA = ${\displaystyle \left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]}$       ...(ii)

From (i) and (ii), we get
AB = BA.