Construct a matrix C as follows: Set A = round(100 rand(4)) L = tril(A,1) + eye(4) C = L

Chapter 7, Problem 3

(choose chapter or problem)

Construct a matrix C as follows: Set A = round(100 rand(4)) L = tril(A,1) + eye(4) C = L L_ (a) The matrix C is a nice matrix in that it is a symmetric matrix with integer entries and its determinant is equal to 1. Use MATLAB to verify these claims. Why do we know ahead of time that the determinant will equal 1? In theory, the entries of the exact inverse should all be integers. Why? Explain. Does this happen computationally? Compute D = inv(C) and check its entries, using format long. Compute C D and compare it with eye(4). (b) Set r = ones(4, 1) and b = sum(C_ ) _ In exact arithmetic the solution of the system Cx = b should be r. Compute the solution by using \ and display the answer in format long. How many digits of accuracy were lost? We can perturb the system slightly by taking e to be a small scalar, such as 1.0e12, and then replacing the right-hand side of the system by b1 = b + e [1,1, 1,1]_ Solve the perturbed system first for the case e = 1.0e12 and then for the case e = 10e06. In each case, compare your solution x with the original solution by displaying x1. Compute cond(C). Is C ill conditioned? Explain.