Question

If $a, b$ are two numbers such that $(a+b)^2 = a^2 + b^2$, then $a.b = 0$.

Is there a similar statement for square matrices.

"If $A, B$ are square matrices such that $(A+B)^2 = A^2 + B^2$, then $A.B = 0$."

Note that if $(A+B)^2 = A^2 + B^2$, then $AB = -BA$, hence $tr(AB) = 0$

Answer 1

If $$ A=\begin{pmatrix} 0&M\\ M&0\end{pmatrix},\qquad B =\begin{pmatrix}-I&0\\ 0&I\end{pmatrix} $$ then $AB+BA=0$, so $A^2+B^2=(A+B)^2$ and neither $A$ nor $B$ is zero.