why can't we use binomial theorem to expand this expression

Question

I was recently told that while expanding expressions of the form

$$(A+B)^n$$

where A and B are square matrices of same order and n is a natural number then

$$(A+B)^n =$$ $$ {{n}\choose{0}}A^{0}B^{n} +{{n}\choose{1}}A^{1}B^{n-1} +....+{{n}\choose{n}}A^{n}B^{0}$$

can be done only if $$AB=BA$$ holds true.

I don't know why it's not true in general and think that $(A+B)^2=A^2+B^2+AB+BA $ might hold some clue. kindly help me out.

Answer 1

You have the answer already, it seems to me. You know that $$(A+B)^2=A^2+B^2+AB+BA\tag1$$ If the proposed formula is true in the $n=2$ case then $$(A+B)^2=A^2+B^2+2AB\tag2$$ Comparing $(1)$ and $(2)$ we see $$AB=BA$$

Doesn't this answer the question?