(i) $\left[\begin{array}{cc}a & b \\ -b & a\end{array}\right]\left[\begin{array}{cc}a & -b \\ b & a\end{array}\right]=\left[\begin{array}{cc}a^2+b^2 & 0 \\ 0 & b^2+a^2\end{array}\right]$.
(ii) $\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]\left[\begin{array}{lll}2 & 3 & 4\end{array}\right]=\left[\begin{array}{lll}1 \times 2 & 1 \times 3 & 1 \times 4 \\ 2 \times 2 & 2 \times 3 & 2 \times 4 \\ 3 \times 2 & 3 \times 3 & 3 \times 4\end{array}\right]$
$$
=\left[\begin{array}{ccc}
2 & 3 & 4 \\
4 & 6 & 8 \\
6 & 9 & 12
\end{array}\right]
$$
(iii) $\left[\begin{array}{cc}1 & -2 \\ 2 & 3\end{array}\right]\left[\begin{array}{lll}1 & 2 & 3 \\ 2 & 3 & 1\end{array}\right]=\left[\begin{array}{ccc}-3 & -4 & 1 \\ 8 & 13 & 9\end{array}\right]$
(iv) $\left[\begin{array}{lll}2 & 3 & 4 \\ 3 & 4 & 5 \\ 4 & 5 & 6\end{array}\right]\left[\begin{array}{ccc}1 & -3 & 5 \\ 0 & 2 & 4 \\ 3 & 0 & 5\end{array}\right]=\left[\begin{array}{ccc}14 & 0 & 42 \\ 18 & -1 & 56 \\ 22 & -2 & 70\end{array}\right]$
(v) $\left[\begin{array}{cc}2 & 1 \\ 3 & 2 \\ -1 & 1\end{array}\right]\left[\begin{array}{ccc}1 & 0 & 1 \\ -1 & 2 & 1\end{array}\right]$
$$
=\left[\begin{array}{ccc}
2 \times 1+1 \times(-1) & 2 \times 0+1 \times 2 & 2 \times 1+1 \times 1 \\
3 \times 1+2 \times(-1) & 3 \times 0+2 \times 2 & 3 \times 1+2 \times 1 \\
-1 \times 1+1 \times(-1) & -1 \times 0+1 \times(2) & -1 \times 1+1 \times 1
\end{array}\right]
$$
$$
=\left[\begin{array}{ccc}
2-1 & 2 & 2+1 \\
3-2 & 4 & 3+2 \\
-1-1 & 2 & -1+2
\end{array}\right]\left[\begin{array}{ccc}
1 & 2 & 3 \\
1 & 4 & 5 \\
-2 & 2 & 0
\end{array}\right]
$$
(vi) $\left[\begin{array}{ccc}3 & -1 & 3 \\ -1 & 0 & 2\end{array}\right]\left[\begin{array}{cc}2 & -3 \\ 1 & 0 \\ 3 & 1\end{array}\right]$
$$
=\left[\begin{array}{cc}
3 \times 2+(-1) \times 1+3 \times 3 & 3 \times(-3)+(-1) \times 0+3 \times 1 \\
-1 \times 2+0 \times 1+2 \times 3 & -1 \times(-3)+0 \times 0+2 \times 1
\end{array}\right]
$$
$$
=\left[\begin{array}{cc}
14 & -6 \\
4 & 5
\end{array}\right] \text {. }
$$