28.2-3 ideas

I think given a matrix multiplication algorithm in M(n), using the previous problem 28.2-2, the LUP decomposition is in O(M(n)) time, then finding the determinant breaks down into finding the determinant of each of L, U, P which is trivial.

I'm not sure about the other direction, that is given determinant algorithm in D(n). Maybe something similar to Cramer's rule for finding the inverse of a matrix, but this procedure is recursive and inefficient.