Matrix multiplication: What is $\mathbf A^3$ and $\mathbf A^n$? Suppose there is matrix A.
I know that A2 = A $\cdot $A
But what if it is A3?
Is it A $\cdot $A $\cdot$A OR A2 $\cdot$ A OR A  $\cdot$ A2?
So basically my question is what is An?
 A: $A^3 = A(AA)=AA^2=(AA)A=A^2A$. This comes from the associativity of matrices. In the case of $A^n =A \cdot A\cdot ...\cdot A$, $n$-times
A: Matrix multiplication is associative, so it is all those things.
A: You doubt is very common with people without a course in abstract algebra.
It doesn't matter if you calculate $(A\cdot A)\cdot A$ or $A\cdot (A\cdot A)$. This is true because the associativity holds for matrices multiplication. Note that you can write the multiplication without parentheses just if you have $(A\cdot A)\cdot A=A\cdot (A\cdot A)$.
Let's take some examples:
Products in the real numbers are associative, for example $(20\cdot 4)\cdot 2 =20\cdot (4\cdot 2)$, because $(20\cdot 4)\cdot 2=160$ and $20\cdot (4\cdot 2)=160$. Then the multiplication is well-defined in this example and you can write $20\cdot 4\cdot 2=160$.
Division are not associative, for example $(20\div 4)\div 2\neq 20\div(4\div2)$, because $(20\div 4)\div 2=2,5$ and $20\div(4\div2)=10$. So   $20\div 4\div 2$ is not well-defined because it would have two results $10$ and $2,5$.
Any questions I would be glad to help.
