In Friedberg's Linear Algebra, the author points out that the evaluation of the determinant of an $n\times n$ matrix by cofactor expansion along any row requires over $n!$ multiplications, whereas evaluating the determinant of an $n\times n$ matrix by elementary row operations can be shown to require only $(n^3+2n-3)/3$ multiplications.
I cannot even figure out the case when $n=2$. The key point, I think, is the exact algorithm of evaluating by elementary row operations. But I temporarily have no idea how to go on.
Here is my question:
How can I deduce the number $(n^3+2n-3)/3$?