TITLE: "Tricks" for solving the determinant of a matrix
QUESTION [15 upvotes]: I've got an exam coming up, and I'm worried that (as I usually tend to do) I will mess up with some minor algebraic calculation in a large problem like finding the determinant of a 4x4 or larger matrix. I'm likely to be asked to use Laplace Expansion, and I'd like to know if there are any other quick and dirty ways I can try to make sure my answer was correct. Considering I'm not alowed to use a calculator, is there some other non-standard method I can use to try to verify a correct solution?
I know how to find the determinant no problem, but I'm very prone to minor mistakes and I don't want to have to spend too much time reading over every problem in fine detail to make sure I didn't omit a negative sign somewhere.

REPLY [2 votes]: You might consider Pivotal Condensation.
PC reduces an $n\times n$ determinant to an $(n-1)\times(n-1)$ determinant whose entries happen to be $2\times 2$ determinants. Simply iterate until your determinant gets to reasonable size. (You can/should stop at $3\times 3$, at which point it's easy enough to compute the final result manually.)
Web-searching "pivotal condensation" will give you lots of explanations and examples. This document even includes a simple proof. This lengthy YouTube video purports to talk through the process, but I didn't view the whole thing.
Pivotal condensation can be extremely tedious; it may-or-may-not be time-effective in an exam situation. However, at every stage of the process, you compute mere $2\times 2$ determinants, which are super-easy and super-quick. So there's a trade-off. (Of course, there's another trade-off: with so many to compute, the danger of stupid arithmetic mistakes accumulates.)
I've actually found pivotal condensation helpful with certain symbolic matrices (in particular, ones with trigonometric terms), since it gives me a systematic way to (attempt to) combine elements. In that regard, it's one of my favorite "tricks".