a) How many multiplications are necessary to find the determinants of matrices which are 2x2, 3x3, 4x4?
b) The number of multiplications for an nxn matrix may be found in terms of the number for an (n-1)x(n-1) matrix. FIND THIS FORMULA and use it to obtain the number of multiplications for a 10x10 matrix.
c) For an nxn matrix the number of multiplications is roughly proportional to n! when n is sufficiently large - find the constant of proportionality from your numerical results from part a.
d) Hence find out approximately how many multiplications and divisions (operations) are needed to use Cramer's rule to solve the system Ax=b where A is nxn.
e) How long will a computer take to do such a computation for a 15x15 matrix if it uses 1 secound to execute 10^6 operations? Compare answer to with time taken using Gaussian Elimination for which the operation count is n(n^2+3n-1)/6.
Gaussian Elimination and Calculation Time are investigated.