Let \((\mathrm{r} \times \mathrm{s})\) denote a matrix with shape $(\mathrm{r} \times \mathrm{s})$. Find the shape of the following products if the product is defined (i) \((2 \times 3)(3 \times 4)\) (iii) \((1 \times 2)(3 \times 1)\) (v) \((3 \times 4)(3 \times 4)\) (ii) \((4 \times 1)(1 \times 2)\) (iv) \((5 \times 2)(2 \times 3)\) (vi) \((2 \times 2)(2 \times 4)\)

Find (i) \(A^{2}\) (ii) \(A^{3}\) (iii) \(A^{4}\) when $A=\begin{array}{rl}1 & 2 \mid \\ & \mid-1\end{array}$

Show that in matrix arithmetic we can have the following: (a) \(\quad \mathrm{AB} \neq \mathrm{BA}\). (b) \(\quad \mathrm{A} \neq 0, \mathrm{~B} \neq 0\) and yet, \(\mathrm{AB}=0\). (c) \(\quad A \neq 0\) and \(A^{2}=0\). (d) \(\quad \mathrm{A} \neq 0, \mathrm{~A}^{2} \neq 0\) and \(\mathrm{A}^{3}=0\). (e) \(\mathrm{A}^{2}=\mathrm{A}\) with \(\mathrm{A} \neq 0\) and $\mathrm{A} \neq \mathrm{I}$. (f) \(\quad A^{2}=\) I with \(A \neq I\) and \(A \neq-I\).

a) If \(\mathrm{A}=\left(\mathrm{a}_{\mathrm{ij}}\right)\) is a $\mathrm{p} \times \mathrm{q}\( matrix and \)\mathrm{B}=\left(\mathrm{b}_{i j}\right)$ is a \(\mathrm{q} \times \mathrm{r}\) matrix prove \(A B\) is the \(p \times r\) matrix \(\left(c_{i j}\right)\) where $c_{i j}={ }^{q} \sum_{k=1} a_{i k} b_{k j}, \quad i=1,2, \ldots, p$ \(\mathrm{J}=1,2, \ldots, \mathrm{r}\) b) If $\mathrm{A}=\begin{array}{ccccc} & 2 & 1 & 1 \mid & \mid 2 & 1 \mid \\\ & \mid-1 & 2 & 3 \mid & \text { and } B=1-1 & 1 \mid, \text { find } A B \text { . } \\ & \mid 1 & 0 & 1 \mid & \mid 2 & -1 \mid\end{array}$

If $\mathrm{A}=\begin{array}{cccccc}2 & -2 & 4 \mid & \text { and } & \mathrm{B}=10 & 1 & -3\end{array} \mid$ \(\mid-1 \quad 1\) \(1 \mid\) 113 find \(2 \mathrm{~A}+\mathrm{B}\).

Prove \((\mathrm{AB}) \mathrm{C}=\mathrm{A}(\mathrm{BC})\) where $\mathrm{A}=\mid 5 \quad \begin{array}{ccc}12 & 2 & 3 \mid \\ -3 & 4 \mid,\end{array}$ $\begin{array}{rlrr}\mathrm{B}= & 12 & -1 & 1 & 0 \\ 10 & 2 & 2 & 21 \\ 13 & 0 & -1 & 31\end{array}$ and $\quad C=\begin{array}{ccc}11 & 0 & 2 \mid \\ 12 & -3 & 0 \mid \\ 12 & 1 & 0 \mid\end{array}$