MATH 2360-002 Exam I July 18, 2008Form B - MakeupAnswer the problems on separate paper. You do not need to rewrite the problem statements on youranswer sheets. Work carefully. Do your own work. Show all relevant supporting steps!1. (12 pts) Each of the following augmented matrices is in row echelon form. A. For each case, indicate whether the corresponding system of linear equations isconsistent or inconsistentB. For each case in which the corresponding system of linear equations is consistent,indicate whether the system has a unique solution or infinitely many solutions.C. For each case in which the corresponding system of linear equations is consistent andhas a unique solution, find that unique solution.a. b. c.11 1201 2100 1300 00⎡⎤⎢⎥−−⎢⎥⎢⎥−−⎢⎥⎢⎥⎢⎥⎣⎦110120102100010⎡⎤−−⎢⎥−−⎢⎥⎢⎥⎣⎦111001 2100 1200 01⎡⎤⎢⎥−−⎢⎥⎢⎥−⎢⎥−⎢⎥⎢⎥⎣⎦2. (10 pts) Each of the following augmented matrices is in reduced row echelon form. For eachcase, find the solution set of the corresponding system of linear equations.a. b.100201010013⎡⎤−⎢⎥−⎢⎥⎢⎥⎣⎦10 20201 10100 01100 000⎡⎤⎢⎥⎢⎥⎢⎥−−⎢⎥⎢⎥⎢⎥⎣⎦3. (10 pts) Consider the following system of linear equations. A. Construct an augmented matrix to represent the system of linear equations.B. Use Gaussian elimination to transform the augmented matrix to a matrix in rowechelon form. State explicitly the specific elementary row operation which is beingdone at each step of the Gaussian elimination.C. Do NOT solve the system of equations.13412 3424124 022xxxxx xxxx−− =⎧⎪−− + =⎨⎪−=−⎩4. (10 pts) Consider the matricesA = B = C = D = 110012−⎡⎤⎢⎥−⎣⎦13221 1−⎡⎤⎢⎥−⎣⎦1211−⎡⎤⎢⎥−⎣⎦2011⎡⎤⎢⎥−−⎣⎦A. For each of the following operations, indicate whether it is possible or not.B. For each of the following operations which is possible, perform it.a. 3A - 2B b. AD c. DC d.TAC5. (8 pts) Let . Find matrices B and C such that and neither is21A63−⎡⎤=⎢⎥−⎣⎦22×BC≠the zero matrix for which the matrix equation AB = AC holds.6. (8 pts) For each of the following pairs of matrices find an elementary matrix E such that EA= B.a.12A31−⎡⎤=⎢⎥−⎣⎦21B31−−⎡⎤=⎢⎥−⎣⎦b.10 2A111212−⎡⎤⎢⎥=−−⎢⎥⎢⎥−−⎣⎦10 2B212111−⎡⎤⎢⎥=−−⎢⎥⎢⎥−−⎣⎦7. (10 pts) Using an augmented matrix, find the inverse of the matrix .11 2A111212−−⎡⎤⎢⎥=−⎢⎥⎢⎥−−⎣⎦8. (12 pts) Find the determinant of each of the following matricesa. b.12A36−⎡⎤=⎢⎥−⎣⎦11 3B211112−⎡⎤⎢⎥=−−⎢⎥⎢⎥−⎣⎦c.012 113 11C221 110 11−⎡⎤⎢⎥−⎢⎥=⎢⎥−− −⎢⎥−⎣⎦9. (3 pts) For each of the matrices in problem 8 (re-given below), determine whether it issingular or non-singular.a. b.12A36−⎡⎤=⎢⎥−⎣⎦11 3B211112−⎡⎤⎢⎥=−−⎢⎥⎢⎥−⎣⎦c.012 113 11C221 110 11−⎡⎤⎢⎥−⎢⎥=⎢⎥−− −⎢⎥−⎣⎦10. (9 pts) Let A and B be matrices such that det(A) = 3 and det(B) = 5. Find the value of 44×a. det(BA) b. det(2B) c. det( )3B11. (8 pts) Find all values of c for which the following matrix is singular11