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

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


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