Let U be an n n upper triangular matrix with nonzeros on the diagonal. Consider solving the system of linear equations U x = b, where b Rn is given and x Rn is the unknown. (a) Write down the last (nth) equation of the system U x = b explicitly, and argue that this equation uniquely determines x(n) (last entry of n). (b) Assuming x(n) has been computed as in part (a), argue that x(n 1) is uniquely determined by the second-to-last ((n 1)st) equation of the system. (c) Proceeding by reverse induction, argue that for k = n, n 1, . . . , 1, all entries of x are uniquely determined. (d) Where did you use the assumption that U has nonzero entries on its diagonal in answering (a)-(c)? Does the argument in (a)-(c) still work in the case that U has one or more zeros on the diagonal