(I'm interested in arbitrary commutative rings, not just integral domains or PID's.)
Let $R$ denote a commutative ring and $M$ denote an $n\times n$ matrix over $R$. Suppose we're interested in claims of the following form:
Proposition. If $\mathrm{det}(M)$ satisfies [blah], then $M$ satisfies [blurg].
Sometimes, such claims are easy to prove, just using the basic properties of determinants. For example, it is easy to see that if $\mathrm{det}(M)$ satisfies "I am not a unit," then $M$ satisfies "I have neither a left inverse, nor a right inverse." The converse is much harder. Explicitly, we're trying to show that: If $\mathrm{det}(M)$ satisfies "I am a unit", then $M$ satisfies "I have a two-sided multiplicative inverse." As I understand it, this can be proved using adjugates. But the underlying insight here is non-trivial, and so too is the proof.
Okay. Lets specialize slightly. Suppose we're interested in claims like:
Proposition. If $\mathrm{det}(M)$ satisfies [generalization of being a unit], then $M$ satisfies [generalization of having a two-sided inverse].
For example: being left-cancellative, two-sided cancellative, non-nilpotent, etc.
Question. What techniques are available to establish such claims?
The proof that $\mathrm{det}(M)$ being unit implies that $M$ has a two-sided inverse is not hard: in this case, the inverse is $\mathrm{det}(M)^{-1} M^{\#},$ where $M^{\#}$ denotes the adjugate. The fact that $M M^{\#} = M^{\#} M = \mathrm{det}(M) I$ is another way of saying that the determinant can be calculated by cofactor expansion, and this formula is valid over any commutative ring.
$M$ being surjective implies that there is a matrix $N$ such that $MN = I$, so $\mathrm{det}(M) \mathrm{det}(N) = 1$ and $\mathrm{det}(M)$ is invertible.
$M$ being injective is equivalent to $\mathrm{det}(M)$ not being a zero-divisor; this is a little harder to prove.
$M$ being non-nilpotent cannot be detected by the determinant. Examples with real coefficients will show you this.
My favorite reference for these topics is chapter VIII of McCoy - Rings and ideals.