We have, $\mathrm{A}=\left[\begin{array}{cc}1 & 2 \\ -2 & 1\end{array}\right], \mathrm{B}=\left[\begin{array}{cc}2 & 3 \\ 3 & -4\end{array}\right]$ and $C=\left[\begin{array}{cc}1 & 0 \\ -1 & 0\end{array}\right]$
(i) $(\mathrm{AB})=\left[\begin{array}{cc}1 & 2 \\ -2 & 1\end{array}\right]\left[\begin{array}{cc}2 & 3 \\ 3 & -4\end{array}\right]=\left[\begin{array}{cc}8 & -5 \\ -1 & -10\end{array}\right]$ and $(\mathrm{AB}) \mathrm{C}=\left[\begin{array}{cc}8 & -5 \\ -1 & -10\end{array}\right]\left[\begin{array}{cc}1 & 0 \\ -1 & 0\end{array}\right]=\left[\begin{array}{cc}13 & 0 \\ 9 & 0\end{array}\right]$
$$
\text { Again, }(\mathrm{BC})=\left[\begin{array}{cc}
2 & 3 \\
3 & -4
\end{array}\right]\left[\begin{array}{cc}
1 & 0 \\
-1 & 0
\end{array}\right]=\left[\begin{array}{cc}
-1 & 0 \\
7 & 0
\end{array}\right]
$$
$$
\begin{aligned}
&\text { and } A(B C)=\left[\begin{array}{cc}
1 & 2 \\
-2 & 1
\end{array}\right]\left[\begin{array}{cc}
-1 & 0 \\
7 & 0
\end{array}\right]=\left[\begin{array}{cc}
13 & 0 \\
9 & 0
\end{array}\right] \\
&\therefore(\mathrm{AB}) \mathrm{C}=\mathrm{A}(\mathrm{BC}) \quad \text { [using Eqs. (i) and (ii) }]
\end{aligned}
$$
$$
\begin{aligned}
&\text { and } \mathrm{A} \cdot(\mathrm{B}+\mathrm{C})=\left[\begin{array}{cc}
1 & 2 \\
-2 & 1
\end{array}\right]\left[\begin{array}{cc}
3 & 3 \\
2 & -4
\end{array}\right]=\left[\begin{array}{cc}
7 & -5 \\
-4 & -10
\end{array}\right] \\
&\text { Also, } \mathrm{AB}=\left[\begin{array}{cc}
1 & 2 \\
-2 & 1
\end{array}\right] \cdot\left[\begin{array}{cc}
2 & 3 \\
3 & -4
\end{array}\right]=\left[\begin{array}{cc}
8 & -5 \\
-1 & -10
\end{array}\right] \\
&\text { and } \mathrm{AC}=\left[\begin{array}{cc}
1 & 2 \\
-2 & 1
\end{array}\right]\left[\begin{array}{cc}
1 & 0 \\
-1 & 0
\end{array}\right]=\left[\begin{array}{ll}
-1 & 0 \\
-3 & 0
\end{array}\right] \\
&\therefore \quad \mathrm{AB}+\mathrm{AC}=\left[\begin{array}{cc}
7 & -5 \\
-4 & -10
\end{array}\right]
\end{aligned}
$$
From Eqs. (iii) and (iv),
$$
\mathrm{A}(\mathrm{B}+\mathrm{C})=\mathrm{AB}+\mathrm{AC}
$$