# Prove that matrix $A$ is symmetric

If $$A$$, $$B$$ are square matrices, $$B$$ is symmetric and $$(A+B)^2$$ is symmetric, prove that $$A$$ is also symmetric.

• Show your effort. Where did you get stuck? – mwt Jan 12 at 14:11
• i used the equality (A+B)^2=((A+B)^2)^T , and B= (B)^T – Kostas Giatzo Jan 12 at 14:40
• You were on the right track, see computation details below. – ex.nihil Jan 12 at 14:43
• Where did you find this exercise? As you can see, the statement is false. – egreg Jan 12 at 15:36
• Guys thank you for the help and your time:) I guess my teacher did a typo in or smth in this one – Kostas Giatzo Jan 13 at 6:19

The statement you want to prove is wrong. Take $$A=\begin{pmatrix} 0 & 1 \\ 0 & 0\end{pmatrix}$$ and $$B$$ the zero matrix. Then $$B$$ is symmetric, $$(A+B)^2 = A^2$$ is the zero matrix (and therefore symmetric), but $$A$$ is not symmetric.