Binomial formula for matrices and commutative matrices

Question

In this post it's said that in order to use the binomial formula to 2 matrices A and B : $(A+B)^{n}$

the two matrices must be commutative. To me , it's sufficient that only one matrix is commutative. Is this true ?

Answer 1

Let $A+B=C$. As addition is always commutative over matrices, $B+A=C$ also.

So $(A+B)^n=C^n=(B+A)^n$, so $AB=BA$, i.e. $A$ and $B$ must be multiplicably commutative for the binomial theorem to provide a unique result.

Commutative is a property of a binary operation, and thus takes two operands to function properly.