General version of Cramer's rule

Question

Let $M$ be a finitely generated algebra over a ring $R$ and let $\{s_1,\dots, s_n\}$ be a generating set. An element $s\in M$ defines a linear transformation via $$s(s_i)=\sum_{j=1}^{n}a_{i,j}s_j$$ Define $A\in M_n(R)$ by $A=(a_{i,j})$. Then the following holds: $$s\pmatrix{s_1\\s_2\\\vdots\\s_n}=A\pmatrix{s_1\\s_2\\\vdots\\s_n}$$ or $$(sI_n-A)\pmatrix{s_1\\s_2\\\vdots\\s_n}=0$$ I have seen a claim that from the equality above it follows that $det(sI_n-A)=0$ and that this is actually a more general version of Cramer's rule. I don't see why.