Describe $n$ x $n$ matrix $A$ , such that it follows $A+A^T = 2A^{-1}$? $A+A^T = 2A^{-1}$ => $(A+A^T)A= 2I$
$$\sum_{k=1}^n a_{kj}(a_{ik}+a_{ki}) = 2I_{ij}$$
So if $i = j$:
$$\sum_{k=1}^na_{ki}(a_{ik}+a_{ki}) = 2$$
And if $i != j$:
$$\sum_{k=1}^n a_{kj}(a_{ik}+a_{ki}) = 0$$
I am now stuck at this stage and can't figure out how to simplify the conditions.

Any symmetric and involutory matrix $B$ will fulfill the identity $B+B^{T}=2B^{-1}$. For example, the first Pauli matrix: $ \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$.
