Why does a matrix PDE $ \hat{M}_{ij}(t) v_j(t) = 0$ imply $| \hat{M}_{ij}(t) | (v_1(t)+v_2(t)+\dots) = 0$?

Question

In a recent paper by Pogorui et al (2021) found here (see the lines leading into eq. 3), it is asserted that the matrix PDE

Implies the following PDE for the sum $f(x,t)=f(x,0,t)+f(x,1,t)$:

Question:

Why is it that the matrix equation being equal to zero implies that the determinant of the matrix multipled by the sum of the vector elements equals zero?

Bonus Question: How does this generalize when the matrix elements $M_{11}$ and $M_{22}$ do not commute?