Matlab Linear Algebra. Please solve the following in Matlab and inculde comments as to what the code is doing. Thank you!
The condition number of a matrix A, defined by cond A A 121IA 2, 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 - H-1b. - 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., ?-??*- 2 and print ? to the screen. Print also the condition number (command: cond CH)) 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. The condition number of a matrix A, defined by cond A A 121IA 2, 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 - H-1b. - 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., ?-??*- 2 and print ? to the screen. Print also the condition number (command: cond CH)) 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
