Why are finding determinants important?
Aside from Cramer's rule when are determinants used?
Determinants appear often when you are interested in finding the inverse of an operator that is a square matrix. They also appear in tensor calculus, which is the mathematical form in which the General Theory of Relativity is written.
In my 50 years as an applied physicist, I have sometimes had to read texts and articles that used determinants in derivations, but I have only had to actually calculate them when using Cramer's Rule to solve a system of equations.
Sometimes when doing detailed solutions of fluid mechanics problems with viscosity, tensor calculus is needed.