Condition for the identity $(a-b)(a+b)=a^2-b^2$

Question

The identity $(a-b)(a+b)=a^2-b^2$ holds true for what condition ???? I tried using real numbers and imaginary numbers but seems like it holds everywhere. Some say... only for ($a>b$) but I don't think so.

Answer 1

It holds for all $a, b$ for which it is true that $ab = ba$ (where multiplication is commutative.)

To see the significance of commutativity, we consider matrix multiplication. In general, the identity is not true when $a, b$ are square matrices of the same dimension, because matrix multiplication is not commutative. In that case $$(a + b)(a -b) = a^2 - ab + ba - b^2$$ Since for matrices in general, $ba \neq ab,$ we do not have $\,-ab + ba \neq -ab + ab$.