What is wrong with this proof that anti-commutative matrices have product zero?

Question

Suppose $AB-BA=0$, where $A$ and $B$ are $n\times n$ matrices. Then $AB=-BA$. Then \begin{aligned} AB=\frac{1}{2}(AB+AB)=\frac{1}{2}(AB-BA)=\frac{1}{2}(0)=0. \end{aligned} I can't seem to find the error in this argument, although I know it's wrong because I found a counterexample by google search.

Answer 1

$AB-BA=0$ gives $AB=BA$, not $AB=-BA$. The $-BA$ term should change sign when you move it across the $=$.