A nifty trick I learned yesterday: suppose you have a big matrix with integer elements, like this one:
and you want to prove that it’s invertible. What do you do? You consider the same matrix modulo 2:
now you can see that it’s triangular and therefore , so the determinant is definitely non-zero, and really is invertible. If it turned out that , it wouldn’t imply that is degenerate, it would mean that the results are inconclusive, and you could, for example, try doing the same thing modulo 3, and so on.