Question

Give examples of matrices
(i) A and B such that AB ≠ BA
(ii) A and B such that AB = O but A ≠ 0, B ≠ 0.
(iii) A and B such that AB = O but BA ≠ O.
(iv) A, B and C such that AB = AC but B ≠ C, A ≠ 0.

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right)\mathrm{Let}A=\left[\begin{array}{cc}1& -2\\ 3& 2\end{array}\right]\mathrm{and}B=\left[\begin{array}{cc}2& 3\\ -1& 2\end{array}\right] \\ AB=\left[\begin{array}{cc}1& -2\\ 3& 2\end{array}\right]\left[\begin{array}{cc}2& 3\\ -1& 2\end{array}\right] \\ \Rightarrow AB=\left[\begin{array}{cc}2+2& 3-4\\ 6-2& 9+4\end{array}\right] \\ \Rightarrow AB=\left[\begin{array}{cc}4& -1\\ 4& 13\end{array}\right] \\ \mathrm{Now}, \\ BA=\left[\begin{array}{cc}2& 3\\ -1& 2\end{array}\right]\left[\begin{array}{cc}1& -2\\ 3& 2\end{array}\right] \\ \Rightarrow BA=\left[\begin{array}{cc}2+9& -4+6\\ -1+6& 2+4\end{array}\right] \\ \Rightarrow BA=\left[\begin{array}{cc}11& 2\\ 5& 6\end{array}\right] \end{gathered}$$

Thus, AB ≠ BA.

$$\begin{gathered} \left(\mathrm{ii}\right)\mathrm{Let}A=\left[\begin{array}{cc}0& 2\\ 0& 0\end{array}\right]\mathrm{and}B=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right] \\ \therefore AB=\left[\begin{array}{cc}0+0& 0+0\\ 0+0& 0+0\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]=O \end{gathered}$$

Thus, AB = O while A ≠ 0 and B ≠ 0.

$$\begin{gathered} \left(\mathrm{iii}\right)\mathrm{Let}A=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\mathrm{and}B=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right] \\ \therefore AB=O \\ \mathrm{and}BA=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right] \\ \Rightarrow BA=\left[\begin{array}{cc}0+0& 1+0\\ 0+0& 0+0\end{array}\right] \\ \Rightarrow BA=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right] \end{gathered}$$

Thus, AB = O but BA ≠ O.

$$\begin{gathered} \left(iv\right)\mathrm{Let}A=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],B=\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]\mathrm{and}C=\left[\begin{array}{cc}0& 0\\ 0& 2\end{array}\right] \\ \therefore AB=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right] \\ \Rightarrow AB=\left[\begin{array}{cc}0+0& 0+0\\ 0+0& 0+0\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right] \\ \mathrm{and}AC=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]\left[\begin{array}{cc}0& 0\\ 0& 2\end{array}\right] \\ \Rightarrow AC=\left[\begin{array}{cc}0+0& 0+0\\ 0+0& 0+0\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right] \end{gathered}$$

Thus,
AB = AC
But B ≠ C and A ≠ 0.