Multiplication of matrices with itself

Question

If A is a matrix when multiplied by itself yields a matrix $A^2$. Now if we wish to obtain $A^3$, are we going to obtain different matrices if I multiply by this way: $A^2\cdot A$ Or this way: $A\cdot A^2$

This may be very simple for you but I am a beginner

Answer 1

In general, we have

$$A^m\cdot A^n=A^n\cdot A^m = A^{m+n}$$

In your case, $m=2, n=1$.

This is due to matrix multiplication is associative $(AB)C=A(BC)$.