The Lindstrom-Gessel-Viennot Lemma is sometimes useful if you can find a nice enough graph corresponding to your matrix. I guess I wouldn't call this canonical though.
Edit: Well, I don't totally understand the downvote. To illustrate with a nice example (which I think is in the original Gessel-Viennot paper), if one takes $A$ to be the $n\times n$ matrix with $[A]_{i,j}={i+j-2\choose i-1}$, then the lemma provides a very elegant way to prove that the determinant of this matrix is 1, and if we then subtract $1$ from the $(n,n)$-entry, then the determinant is 0.
