If A = [-1112301-31],B=[214302121], state whether AB = BA? Justify your answer.

Exercise 2.3 | Q 2 | Page 55

QUESTION

If A = ${\displaystyle \left[\begin{array}{ccc}-1& 1& 1\\ 2& 3& 0\\ 1& -3& 1\end{array}\right],\text{B}=\left[\begin{array}{ccc}2& 1& 4\\ 3& 0& 2\\ 1& 2& 1\end{array}\right]}$, state whether AB = BA? Justify your answer.

SOLUTION

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

= ${\displaystyle \left[\begin{array}{ccc}-2+3+1& -1+0+2& -4+2+1\\ 4+9+0& 2+0+0& 8+6+0\\ 2-9+1& 1+0+2& 4-6+1\end{array}\right]}$

∴ AB = ${\displaystyle \left[\begin{array}{ccc}2& 1& -1\\ 13& 2& 14\\ -6& 3& -1\end{array}\right]}$     ...(i)

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

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

∴ BA = ${\displaystyle \left[\begin{array}{ccc}4& -7& 6\\ -1& -3& 5\\ 4& 4& 2\end{array}\right]}$      ...(ii)
From (i) and (ii), we get
AB ≠ BA.