We have \begin{equation} \begin{split} \mathsf{tr}(\alpha A + \beta B) & = \sum_{j = 1}^{\min(N, M)} (\alpha a_{i, j} + \beta b_{i, j}) \\ & = \sum_{j = 1}^{\min(N, M)} \alpha a_{i, j} + \sum_{j = 1}^{\min(N, M)} \beta b_{i, j} \\ & = \alpha \sum_{j = 1}^{\min(N, M)} a_{i, j} + \beta \sum_{j = 1}^{\min(N, M)} b_{i, j} \\ \end{split} \end{equation} $\square$