We know from class that computing the determinant of an n n matrix using LU decomposition has computational cost W = O(n^3). But how expensive is it to compute a determinant using the common recursive definition formula for determinants?

(a) Determine the number of floating point operations required for calculating the determinant of an n n matrix using a straightforward implementation of the recursive definition of the determinant. (You can combine additions and multipli- cations.) Hint: consider the number of flops done on each recursive level separately. How many determinants are to be computed on each recursive level? On each recursive level, how many flops does it take to compute each of these determinants? (By us iug determinauis thai are o: iza: saualler, omulo th rexve level Then sum up over all the levels (give the result as a sum like -2 ). (b) Find an approximation for the expression derived that is valid for large n. (Hint this approximate result should be a very simple expression, proportional to n!, and you will need expz 12/3! /4!+