Fix $n \in \{ 1, \ldots, N \}$ and $m \in \{ 1, \ldots, M \}$. Then \begin{equation} (A + B)^T_{n, m} = (A + B)_{m, n} = A_{m, n} + B_{m, n} = A^T_{n, m} + B^T_{n, m} \end{equation} $\square$