Finding the determinant of an $n\times n$ matrix.... and the inverse

Question

Finding the determinant of a $2\times 2$ matrix is easy and the inverse is even easier. Finding the determinant of a $3\times 3$ matrix and its inverse is a little more difficult but still doable. $4\times 4$ I can use RREF of an augmented identity and deduce the determinant from that. But as the size of the square matrix grows, finding these two becomes a serious PITA. What I was wondering is whether or not there is a more efficient method to find the determinant of an arbitrarily large $n\times n$ matrix by hand..... just because.

Answer 1

The "brute force" method would require $n!$ operations, but if you bring the matrix into upper triangular form by row operations and then read the determinant from th eproduct of the diagonal entries, it only take $O(n^3)$ operations.