Give sufficient details of your working to make it clear that you have not
used any software or calculator that can perform matrix manipulation. For
example, in part (a), show explicitly how you calculated at least one element
of each matrix answer.
(a) Let
$$A = \begin{pmatrix} 2 & 3 \\ -1 & 4 \\ 0 & -2 \end{pmatrix}, B = \begin{pmatrix} 4 & -1 \\ 6 & 2 \end{pmatrix}, \text{and} C = \begin{pmatrix} 3 & -1 & 4 \\ -2 & 5 & 7 \end{pmatrix}$$
Evaluate each of the following expressions, if possible. Where evaluation
is not possible, explain why not.
(i) AB
(ii) BA
(iii) B^2
(iv) CA
(v) 4B - 2CA.
(b) Determine whether the following matrices are invertible, and find the
inverse for each case that it exists.
$$
\begin{aligned}
&(i) \begin{pmatrix} 6 & 10 \\ -3 & -5 \end{pmatrix} \\
&(ii) \begin{pmatrix} 5 & -9 \\ 3 & -6 \end{pmatrix}
\end{aligned}
$$
