In my undergraduate mechanics class, we just finished the section on coupled oscillators where we use eigenvalues to find solutions to the differential equations describing the motion of our spring mass system. When we finished, one of my fellow students started going on a rant about how he "knew the determinant had a physical meaning," and he went on to talk about how frustrated he was with his differential equations professor who said that the determinant was just an operation and only had physical meaning when it was given one (ie- showing the system is linearly independent by setting the determinant to $0$). I sided with the professor, but I wasn't sure how to explain it to my classmate.

What do you guys think?