Let $F,E$ be two locally free sheaves of equal rank on a complex manifold $X$.
Assume that we have an injection $0\to F\to E$ such that the induced map $0\to \det F\to \det E$ is an isomorphism. Is then the original map an isomorphism too? If not always, what if also $\det F=\det E=\mathcal{O}_X$ ?
The question is local, so pass to the stalks, where both $E,F$ can be assumed free.
Then the statement is that a module endomorphism of $R^n$ is an isomorphism iff its determinant is a unit (rather than merely a nonzerodivisor), which is true.