I'm learning Linear Algebra using MIT's Open Courseware Course 18.06
Quite often, the professor says "... assuming that the matrix is invertible ...".
Somewhere in the lecture he says that using a determinant on an $n \times n$ matrix is on the order of $O(n!)$ operations, where an operation is a multiplication and a subtraction.
Is there a more efficient way? If the aim is to get the inverse, rather than just determine the invertibility, what is the most effecient way to do this?