MATH 2360-001 Exam I 28 September 2009Form AAnswer the problems on separate paper. You do not need to rewrite the problem statements on youranswer sheets. Work carefully. Calculators are not permitted. Do your own work. Show all relevantsupporting steps!1. (6 pts) Identify which of the following matrices are (i) in reduced row echelon form [RREF], (ii) in row echelon form but not reduced row echelon form [ROW], (iii) not in row echelon form [NOT]. Use the labels RREF, ROW and NOT to denote your answers. (Note options (i), (ii)and (iii) are mutually exclusive.) a. b. c.12 101 000 0−⎡⎤⎢⎥⎢⎥⎢⎥⎣⎦110001010−⎡⎤⎢⎥⎢⎥⎢⎥⎣⎦10 101 200 0−⎡⎤⎢⎥⎢⎥⎢⎥⎣⎦d. e. f.100120001⎡⎤⎢⎥⎢⎥⎢⎥⎣⎦110001000001−⎡⎤⎢⎥⎢⎥⎢⎥⎣⎦110000010000−⎡⎤⎢⎥⎢⎥⎢⎥⎣⎦2. (3 pts) Each of the following augmented matrices is in row echelon form. For each case, indicate whether the corresponding system of linear equations isconsistent or is inconsistenta. b. c.10 1201 2000 11⎡⎤−−⎢⎥⎢⎥⎢⎥⎣⎦110101020000⎡⎤−⎢⎥−⎢⎥⎢⎥⎣⎦10 1001 2000 01⎡⎤−⎢⎥⎢⎥⎢⎥⎣⎦3. (3 pts) Each of the following augmented matrices from Problem 2 and re-given below is inrow echelon form. 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.a. b. c.10 1201 2000 11⎡⎤−−⎢⎥⎢⎥⎢⎥⎣⎦110101020000⎡⎤−⎢⎥−⎢⎥⎢⎥⎣⎦10 1001 2000 01⎡⎤−⎢⎥⎢⎥⎢⎥⎣⎦4. (3 pts) Each of the following augmented matrices from Problem 2 and re-given below is inrow echelon form. For each case in which the corresponding system of linear equations is consistent andhas a unique solution, find that unique solution.a. b. c.10 1201 2000 11⎡⎤−−⎢⎥⎢⎥⎢⎥⎣⎦110101020000⎡⎤−⎢⎥−⎢⎥⎢⎥⎣⎦10 1001 2000 01⎡⎤−⎢⎥⎢⎥⎢⎥⎣⎦5. (12 pts) Each of the following augmented matrices is in reduced row echelon form. For eachcase, find the complete solution set of the corresponding system of linear equations.a. b.10 1201 1300 00⎡⎤−⎢⎥−⎢⎥⎢⎥⎣⎦100301020011⎡⎤−⎢⎥⎢⎥⎢⎥−⎣⎦6. (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 3424123 02xxxxx xxxx−− =⎧⎪+− + =⎨⎪−=−⎩7. (4 pts) Consider the matricesA = B = C = D = 11 2112−−⎡⎤⎢⎥−⎣⎦1211−⎡⎤⎢⎥−⎣⎦11201 1−⎡⎤⎢⎥−⎣⎦2011⎡⎤⎢⎥−−⎣⎦For each of the following operations, indicate whether it is possible or not.a. 2A - C b. BC c. AD d.TCB8. (12 pts) Consider the matrices given in Problem 7 and re-given belowA = B = C = D = 11 2112−−⎡⎤⎢⎥−⎣⎦1211−⎡⎤⎢⎥−⎣⎦11201 1−⎡⎤⎢⎥−⎣⎦2011⎡⎤⎢⎥−−⎣⎦For each of the following operations which is possible, perform it.a. 2A - C b. BC c. AD d.TCB9. (8 pts) Let . Find matrices B and C such that and neither is the20A10−⎡⎤=⎢⎥⎣⎦22×BC≠zero matrix for which the matrix equation AB = AC holds.10. (8 pts) For each of the following pairs of matrices find an elementary matrix E such thatEA = B.a.12A31−⎡⎤=⎢⎥−⎣⎦48B31−⎡⎤=⎢⎥−⎣⎦b.10 2A111212−⎡⎤⎢⎥=−−⎢⎥⎢⎥−−⎣⎦10 2B111114−⎡⎤⎢⎥=−−⎢⎥⎢⎥−−⎣⎦11. (14 pts) Find the determinant of each of the following matricesa. b.12A24−⎡⎤=⎢⎥−⎣⎦10 2B111012−⎡⎤⎢⎥=−−⎢⎥⎢⎥−⎣⎦c.012 111 13C221 110 11−⎡⎤⎢⎥−⎢⎥=⎢⎥−− −⎢⎥−⎣⎦12. (3 pts) For each of the matrices in Problem 11 and re-given below, determine whether it issingular or non-singular.d. b.12A24−⎡⎤=⎢⎥−⎣⎦10 2B111012−⎡⎤⎢⎥=−−⎢⎥⎢⎥−⎣⎦c.012 111 13C221 110 11−⎡⎤⎢⎥−⎢⎥=⎢⎥−− −⎢⎥−⎣⎦13. (9 pts) Let A and B be matrices such that det(A) = 3 and det(B) = -4. Find the value of 33×a. det(BA) b. det(2B) c. det( )2B14. (8 pts) Find all values of c for which the following matrix is singular10