(1) Use "substitution" to solve L*x=b where
   L = [ 1 0 0 ]  b = [ 3  ]
       [ 2 1 0 ]      [ -4 ]
       [ 4 3 1 ]      [ 0  ]

(2) Use "substitution" to compute L^{-1}, where L is given above.

(3) In lecture we showed that the number of arithmetic operations
needed to  compute an LU decomposition of an n x n matrix with 
rank n was 2/3*n^3 + lower order terms, i.e. terms proportional
to n^2 or n.

  (3a) How many more arithmetic operations does it take,
       given the LU decomposition of A, to compute A^{-1}?
       The method to use is for each column e_j of the 
       identity matrix, solve Ax = e_j for x = column j of A^{-1}.
       Use substitution with the L and U factors to solve this
       equation. Recall that the P_L and P_R factors only involve
       reordering, no arithmetic operations. Your answer should
       be of the form "c1*n^c2 + lower order terms", where
       c1 and c2 are constants you need to determine.
       Add 2/3*n^3 to your answer to determine the total number
       of arithmetic operations to compute the inverse of an 
       invertible n x n matrix.

  (3b) Now suppose A is m x n with rank r (the general case).
       How many arithmetic operations are needed to compute
       the LU decomposition?