Matlab: The condition number of a matrix A, defined by cond A = kAk2kA1k2, allows one to estimate the accuracy of a computed solution of a system Ax = b. If the entries of A and b are accurate to about r significant digits, and cond A k, then the computed solution of Ax = b should usually be accurate to at least r k significant digits.

Solve the following two problems: Compute the Hilbert matrix H of order k = 5 and k = 12 using the hilb command. For k = 5, solve the system Hx = b for a suitable b to find the last column of the inverse of H. Use the backslash command. To find a suitable b think of the solution as x = H1b. For k = 5 and k = 12, solve the system Hx = y using the backslash command, where y is generated by applying H to a random vector x ? , i.e., y = Hx? . Compute the 2-norm of the difference between x ? and x, i.e., = kx ? xk2 and print to the screen. Print also the condition number (command: cond(H)) of H to the screen. Describe what you observe.

Find the determinant (command: det(A)) and the condition number (command: cond(A)) of the Hilbert matrix H of order k (command: hilb(k)), for k = 1, 2, . . . , 10. Plot the determinant and the condition number as a function of k using a logarithmic scale for the vertical axis