If
$A=\left[\begin{array}{cc}1 & 2 \\ -2 & 1\end{array}\right], \quad B=\left[\begin{array}{cc}2 & 3 \\ 3 & -4\end{array}\right]$ and $\quad C=\left[\begin{array}{cc}1 & 0 \\ -1 & 0\end{array}\right]$
(i) (AB) C = A (BC)
(ii) A (B + C) = AB + AC.
Given, $A=\left[\begin{array}{cc}1 & 2 \\ -2 & 1\end{array}\right], B=\left[\begin{array}{cc}2 & 3 \\ 3 & -4\end{array}\right]$ and $C=\left[\begin{array}{cc}1 & 0 \\ -1 & 0\end{array}\right]$
(i) $A B=\left[\begin{array}{cc}1 & 2 \\ -2 & 1\end{array}\right]\left[\begin{array}{cc}2 & 3 \\ 3 & -4\end{array}\right]=\left[\begin{array}{cc}2+6 & 3-8 \\ -4+3 & -6-4\end{array}\right]=\left[\begin{array}{cc}8 & -5 \\ -1 & -10\end{array}\right]$
and $(A B) 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}8+5 & 0 \\ -1+10 & 0\end{array}\right]=\left[\begin{array}{cc}13 & 0 \\ 9 & 0\end{array}\right]$$\ldots$ (i)
Again, $(B C)=\left[\begin{array}{cc}2 & 3 \\ 3 & -4\end{array}\right]\left[\begin{array}{cc}1 & 0 \\ -1 & 0\end{array}\right]=\left[\begin{array}{cc}2-3 & 0 \\ 3+4 & 0\end{array}\right]=\left[\begin{array}{cc}-1 & 0 \\ 7 & 0\end{array}\right]$
And $A(B C)=\left[\begin{array}{cc}1 & 2 \\ -2 & 1\end{array}\right]\left[\begin{array}{cc}-1 & 0 \\ 7 & 0\end{array}\right]=\left[\begin{array}{cc}-1+14 & 0 \\ 2+7 & 0\end{array}\right]=\left[\begin{array}{cc}13 & 0 \\ 9 & 0\end{array}\right]$...(ii)
From (i) and (ii), we get
Hence, $(A B) C=A(B C)$
(ii) $\quad B+C=\left[\begin{array}{cc}2 & 3 \\ 3 & -4\end{array}\right]+\left[\begin{array}{cc}1 & 0 \\ -1 & 0\end{array}\right]=\left[\begin{array}{cc}3 & 3 \\ 2 & -4\end{array}\right]$
Now,
$A \cdot(B+C)=\left[\begin{array}{cc}1 & 2 \\ -2 & 1\end{array}\right]\left[\begin{array}{cc}3 & 3 \\ 2 & -4\end{array}\right]$
$=\left[\begin{array}{cc}3+4 & 3-8 \\ -6+2 & -6-4\end{array}\right]$
$=\left[\begin{array}{cc}7 & -5 \\ -4 & -10\end{array}\right]$ ...(iii)
$A B=\left[\begin{array}{cc}1 & 2 \\ -2 & 1\end{array}\right]\left[\begin{array}{cc}2 & 3 \\ 3 & -4\end{array}\right]=\left[\begin{array}{cc}2+6 & 3-8 \\ -4+3 & -6-4\end{array}\right]=\left[\begin{array}{cc}8 & -5 \\ -1 & -10\end{array}\right]$
and, $A C=\left[\begin{array}{cc}1 & 2 \\ -2 & 1\end{array}\right]\left[\begin{array}{cc}1 & 0 \\ -1 & 0\end{array}\right]=\left[\begin{array}{cc}1-2 & 0 \\ -2-1 & 0\end{array}\right]=\left[\begin{array}{cc}-1 & 0 \\ -3 & 0\end{array}\right]$
Thus, $A B+A C=\left[\begin{array}{cc}8 & -5 \\ -1 & -10\end{array}\right]+\left[\begin{array}{cc}-1 & 0 \\ -3 & 0\end{array}\right]=\left[\begin{array}{cc}7 & -5 \\ -4 & -10\end{array}\right]$...(iv)
Hence from equation (iii) and (iv), we have
$A \cdot(B+C)=A \cdot B+A \cdot C$