TITLE: Matrix P to the power of 4, i.e $P^4$, is this the same as $P^2 \cdot P^2$?
QUESTION [5 upvotes]: Basically, what it says in the title. 
I have a $5 \times 5$ matrix and I need to work out $P^4$, is it possible to just do $P^2$ and multiply this with itself?

REPLY [2 votes]: What does $P^4$ mean to you? It might mean $$P\cdot P\cdot P\cdot P$$ but this is not exactly defined until you agree that matrix multiplication is asociative. Otherwise $P^4$ should mean something like $$((P\cdot P)\cdot P)\cdot P$$ Now if you believe that the multiplication is associative, then this is the same as $$(P\cdot P)\cdot (P\cdot P)$$ which is almost certainly what you would mean by $P^2\cdot P^2$.