Two real matrices are given:

$$A=\begin{pmatrix}
2 & 5 & -3\\ 
0 & -1 & 6
\end{pmatrix} \text{ and } B=\begin{pmatrix}
3 & 1\\ 
4 & -2\\ 
0 & 6
\end{pmatrix}$$

Now I want show that $(A \cdot B)^{2}= (A \cdot B) \cdot (A \cdot B)$

The problem is I don't know if we are allowed to say that both sides are the same. That means I have to do it on both sides I think :(

Or what do you think?

Assuming I have to do it on both sides, how would the left side look then, correctly written as matrices?

Would this notation be correct?
$$(A \cdot B)^{2} = \begin{pmatrix}
2 \cdot 3 + 5 \cdot 4 -3 \cdot 0 & 2 \cdot 1+5(-2) -3 \cdot 6 \\ 
0 \cdot 3 -1 \cdot 4+6 \cdot 0 & 0 \cdot 1-1(-2)+6 \cdot 6
\end{pmatrix}^{2}= \begin{pmatrix}
26 & -26\\ 
-4 & 38
\end{pmatrix}^{2}= \begin{pmatrix}
26 & -26\\ 
-4 & 38
\end{pmatrix} \cdot \begin{pmatrix}
26 & -26\\ 
-4 & 38
\end{pmatrix}= \begin{pmatrix}
26 \cdot 26 -26(-4) & 26(-26)-26 \cdot 38\\ 
-4 \cdot 26+38 \cdot (-4) & -4 \cdot (-26)+38 \cdot 38
\end{pmatrix}= \begin{pmatrix}
780 & -1664\\ 
-256 & 1548
\end{pmatrix}$$

Can I write it like that and do I have to calculate the other side really too?