Let be a square matrix and for that it holds that

Prove that is symmetric.

**Solution**

(1)

Taking transposed matrices back at we get that

and thus

(2)

On the other hand it holds that

since

Of course it holds that if a matrix is symmetric or antisymmetric and then . *The proof is left as an exercise to the reader.*

We can safely conclude using the above observations that for our matrix it holds that

(3)

But then for the matrix it holds that

and since the matrix is antisymmetric we conclude that and thus . Hence the result.