I. (a) Solve Ax b where b 3,5l' and A is below, using Gaussian Elimination with Back Solving. Use pivoting if appropriate. (b) Use your work in (a), to give the LU decomposition of A 2. Write a MATLAB function program that solves the linear systems Axi-bi and Ax2 b using LU decomposition. Let A, bi, and b2 be the inputs and xi and x2 be the outputs. Include comments. 3. Find, using the characteristic polynomial, the eigenvalues and eigenvectors of the matrix . Describe how you would use the inverse power method to find one eigenvalue and eigenvector For B above, which one would it find? 5. Find, using the characteristic polynomial, the eigenvalues and eigenvectors of the matrix 6. Write a MATLAB script program that will use Newton's method to find a root of the system 1 starting from the initial guess of functions ,y)7y1 and a(r, y) (0,0). Include comments. 7. What is the condition number of a matrix? How do you find it with MATLAB? What are the implications of the condition number when solving a linear system? In what form does it often appear in MATLAB warnings? 8. Write a MATLAB function program to that solves a linear system Ax-b using LU decom- position. Let A, b and to be the inputs and x the output. If the error (residual) is not less than tol, then display a warning. 0 2 9. Suppose D Using vo-(1,1 as the starting vector do 2 iterations of the Power Method for D. What do the results mean? 10. Write a MATLAB function program to do n iterations of the Power Method that will work if the eigenvalues of the input matrix are all real and positive. Let the matrix A and n be inputs and let fe v r (the eigenvalue, eigenvector and resal) be the outputs 11. Give the MATLAB commands, or sequences of commands for solving a linear system Ax-b in as many ways as you know. Which of these are the worst and best