18.06 Fall 2004 Quiz 1 October 13, 2004Your name is:Please circle your recitation:1. M2 A. Brooke-Taylor2. M2 F. Liu3. M3 A. Brooke-Taylor4. T10 K. Cheung5. T10 Y. Rubinstein6. T11 K. Cheung7. T11 V. Angeltveit8. T12 V. Angeltveit9. T12 F. Rochon10. T1 L. Williams11. T1 K. Cheung12. T2 T. GerhardtGrading:Question Points MaximumName + rec 51 152 553 25Total: 100Remarks:Do all your work on these pages.No calculators or notes.Putting your name and recitation section correctly is worth 5 points.The exam is worth a total of 100 points.11. LetA =2 2 24 3 1−2 −1 4.(a) Compute an LDU factorization of A if one exists.2(b) Give all solutions to Ax = b where b =2−311.32. One of the entries of A has been modified as there was a mistake. (Many of the subquestionsare independent and can be answered in any order.) By performing row eliminations (and possiblypermutations) on the following 4 × 8 matrix A1 2 0 3 -1 1 1 -2-3 -6 2 -7 7 0 -6 31 2 2 5 3 3 -1 02 4 0 6 -2 1 3 0we got the following matrix B:1 2 0 3 -1 0 2 00 0 1 1 2 0 0 00 0 0 0 0 1 -1 00 0 0 0 0 0 0 1(a) What is the rank of A?(b) What are the dimensions of the 4 fundamental subspaces?4(c) How many solutions does Ax = b have? Does it depend on b? Justify(d) Are the rows of A linearly independent? Why?(e) Do columns 4, 5, 6 and 7 of A form a basis of R4? Why?5(f) Give a basis of N (A).(g) Give a basis of N (AT).6(h) (You do not need to do any calculations to answer this question.) What is the reduced rowechelon form for AT? Explain.(i) (Again calculations are not necessary for this part.) Let B = EA. Is E invertible? If so, whatis the inverse of E?73. For each of these statements, say whether the claim is true or false and give a brief justification.(a) True/False: The set of 3 × 3 non-invertible matrices forms a subspace of the set of all 3 × 3matrices.(b) True/False: If the system Ax = b has no solution then A does not have full row rank.8(c) True/False: There exist n × n matrices A and B such that B is not invertible but AB isinvertible.(d) True/False: For any permutation matrix P , we have that P2=
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