Let $M_1$ and $M_2$ be two square matrices and let $M = \operatorname{diag}(M_1, M_2)$. Prove the relation $\operatorname{dim}\ker M = \operatorname{dim}\ker M_1 + \operatorname{dim}\ker M_2$ and then that for all nonzero integer $k$,
$$\delta_k(M) = \delta_k(M_1) + \delta_k(M_2).$$
One may use without proof the fact that all these relations extend to a block diagonal matrix $\operatorname{diag}(M_1, \ldots, M_s)$.