Q.Give your own examples of matrices satisfying the following conditions in each case:

Step 1. (i) Choose $A,B$ and test commutativity. Take $A=\begin{pmatrix}1&1\\0&1\end{pmatrix},\ B=\begin{pmatrix}1&0\\1&1\end{pmatrix}$.

$AB=\begin{pmatrix}1(1)+1(1)&1(0)+1(1)\\0(1)+1(1)&0(0)+1(1)\end{pmatrix}=\begin{pmatrix}2&1\\1&1\end{pmatrix}$

$BA=\begin{pmatrix}1(1)+0(0)&1(1)+0(1)\\1(1)+1(0)&1(1)+1(1)\end{pmatrix}=\begin{pmatrix}1&1\\1&2\end{pmatrix}$

Since $AB=\begin{pmatrix}2&1\\1&1\end{pmatrix}\ne\begin{pmatrix}1&1\\1&2\end{pmatrix}=BA$, this pair satisfies (i).

Step 2. (ii) Choose $A,B$ nonzero with both products zero. Take $A=B=\begin{pmatrix}0&1\\0&0\end{pmatrix}$ (nonzero).

$AB=\begin{pmatrix}0&1\\0&0\end{pmatrix}\begin{pmatrix}0&1\\0&0\end{pmatrix}=\begin{pmatrix}0(0)+1(0)&0(1)+1(0)\\0&0\end{pmatrix}=\begin{pmatrix}0&0\\0&0\end{pmatrix}=O$.

Since $A=B$ here, $BA=AB=O$ too. Both $A\ne O$ and $B\ne O$, so this pair satisfies (ii).

Step 3. (iii) Choose $A,B$ with $AB=O$ but $BA\ne O$. Take $A=\begin{pmatrix}1&0\\0&0\end{pmatrix},\ B=\begin{pmatrix}0&0\\1&1\end{pmatrix}$. …

• Assuming matrix multiplication is commutative like real-number multiplication …