A. MillerM340 Final ExamMay 14, 1991There is a total of 200 p oints. You can pick up your gradedexam in VanVleck 403. Your grade will be posted on the door.1. (10 points) Find the inverse of A or show that the inverse does not exist.A =0 0 0 −10 0 1 00 −1 0 01 0 0 02. (20 points) Suppose that M and N are invertible matrices such thatMAN = B.a. If H is a basis matrix for C(B), then what matrix is a basis matrix forC(A)?b. If K is a basis matrix for N (A), then w hat matrix is a basis matrixfor N (B)?c. If R is a right inverse of A, then what matrix is a right inverse of B?d. If L is a left inverse of B, then what matrix is a left inve rse of A?3. (20 points) Find a diagonal matrix D and invertible matrix P such thatA = P DP−1whereA ="4 −12 1#4. (10 points) h"1 + i1#,"1 + i3#i=5. (10 points) |2 + 3i|=6. (10 points) ||"i1#||=17. (15 points) Let T : V → W be a mapping.a. Define T is one-to-one.b. Prove that if T is one-to-one, then for every X, Y ⊆ VT (X) ∩ T (Y ) ⊆ T (X ∩ Y ).8. (15 points) Prove: For all U, V ∈ Rn×1|hU, V i| ≤ ||U || ||V ||9. (15 points) Prove that if A ∈ Cn×nis Hermetian (ie. AT= A), then theeigenvalues of A are real.True or false, enter your answers on the answer sheet.10. If a real matrix is both unitary and Hermetian, then it must be theidentity matrix.11. If U and V are n × n uppertriangular matrices then so is UV .12. If D is a diagonal n × n matrix, then so is cDn+1where c is a scalar.13. If A, B, C and D are n × n matrices, then (AB)(CD) = A(BC)D.14. If A and B are n × n matrices and c is a scalar, then A(cB) = B(cA).15. If A, B, and C are n × n matrices and A + C = C + B, then A = B.16. If A is an n×n invertible matrix and B is n×1, then the matrix equationAX = B has a unique solution.17. If U and V are unitary n × n matrices, then UV is a unitary matrix.18. An n × n matrix U is unitary iff for every X which is n × 1 we have||UX|| = ||X||.19. If A and B are Hermetian n × n matrices, then A + B is Hermetian.20. If A, B, and C are n × n matrices and AC = BC, then either A = B orC is the zero matrix.21. It is possible to find 5 × 1 vectors U and V such that ||U||=2, ||V || = 3and ||U + V || = 4.222. It is possible to find 17 × 1 vectors U and V such that ||U||=4, ||V || = 2and ||U + V || = 7.23. It is possible to find 3 × 1 vectors U and V such that hU, V i = 5 and||U|| = ||V || = 2.24. For any n there exists an n × n elementary matrix E such that EA = ATfor any n × n matrix A.25. For any m × n matrix A it is possible to find an invertible m × m matrixM such that MA is in row echelon form.26. The sum of two n × n invertible matrices is invertible.27. If A and B are similar matrices, then trace(A) = trace(B).28. If A and B are similar, then det(A) = det(B).29. If two matrices A and B have the same characteristic polynomial, thenthey are similar.30. If two matrices A and B are similar, then they have the same character-istic polynomial.31. Suppose that A ∈ Fn×m, B ∈ Fm×p, C ∈ Fp×r. Then (AB)C = A(BC).32. If W1and W2are subspaces of Fn×1, then defineW1MW2= {w1+ w2: w1∈ W1, w2∈ W2}.Then W1LW2is a subspace of V .33. Suppose that u1, u2are orthogonal unit vectors in Rn×1and c1, c2∈ R,then ||c1u1+ c2u2|| =qc21+ c22.34. The inverse of a Hermetian matrix is Hermetian.3Answer sheet for true and false.Each one is worth 3 points.10. Circle one: True or False11. Circle one: True or False12. Circle one: True or False13. Circle one: True or False14. Circle one: True or False15. Circle one: True or False16. Circle one: True or False17. Circle one: True or False18. Circle one: True or False19. Circle one: True or False20. Circle one: True or False21. Circle one: True or False22. Circle one: True or False23. Circle one: True or False24. Circle one: True or False25. Circle one: True or False26. Circle one: True or False27. Circle one: True or False28. Circle one: True or False29. Circle one: True or