Pleаse аnswer questiоns 1 аnd 2 оn page 1 and questiоns 3 and 4 on page 2. Let A= . 1. Find all the eigenvalues of A. (7 points) 2. Find the corresponding eigen vectors. (7 points) 3. Can A be diagonalized? (2 points) 4. If the answer to 3 is yes, find a matrix X and a diagonal matrix D so that X^{-1}*A*X is a diagonal matrix D. (4 points) Old Quiz 6: ------------- 1. Use Cramer's rule to find the solutions of the system of equations: x1+ 3x2+ x3=1 2x1+x2+x3 =5 -2x1+2x2 - x3= -8 This can broken into the following subproblems: (i) Compute the determinant of the coefficient matrix for the system of equations above. (5 points) (ii) Compute the determinant of the three matrices obtained by replacing one column of the coefficient matrix by the vector on the right hand side. (4 points each) (iii) Write the solutions for each of the variables x1, x2 and x3. (3 points)
Pleаse fоllоw these instructiоns cаrefully, аs failure to follow them will result in a penalty. (i) Download the Exam template from the class website and use that to write your solutions. (ii) The solutions to each problem should be in the space specially assigned to them. (iii) When scanning your solutions into a pdf file, each page must be scanned as a separate page and the entire exam as one pdf file. (iv) You have 1 hour and 30 minutes to complete the exam, including the time to scan the exam and upload it as a pdf file to Proctorio I. Find all solutions to the system of equations x1 + 3x2 + x3 + x4 = 3 2x1 - 2x2 + x3 + 2x4 = 8 3x1 + x2 + 2x3 - x4 = -1 (20 points) II. Let A denote the coefficient matrix for the system of equations given in I. i) Find a basis for the Column space of A. (10 points) ii) Find a basis for the Row space of A. (5 points) iii) Find a basis for the Null space of A. (10 points) iv) What is the dimension of the Column space of A? What is the dimension of the Null space of A? (5 points) III. Solve the matrix equation A.x=b, where A= , x= and b= by first finding the inverse of the matrix A.(25 points) IV. Write the matrix A in III as a product of elementary matrices. (25 points)