
Let A € M3x2(R) and B € M2x3(R) be matrices satisfying AB= 8 2 -2 2-27...
44. a.Let A and B be two 2 × 2 matrices,Let Tr denote the trace and det denote the determinant. Prove that Tr(AB)-Tr(BA) and det(AB) - det(BA). b. If A is any matrix in SLa(R), prove that det ((-A-t +1 where t = Tr(A).
44. a.Let A and B be two 2 × 2 matrices,Let Tr denote the trace and det denote the determinant. Prove that Tr(AB)-Tr(BA) and det(AB) - det(BA). b. If A is any matrix in SLa(R), prove...
5. Prove or disprove the following statements (a) Let A B and C be 2 x 2 matrices. If AB = AC, then B = C (b) If Bvi,.., Bvh} is a then vi, . ., vk} is a linearly independent set in R". linearly independent set in R* where B is a kx n matrix,
5. Prove or disprove the following statements (a) Let A B and C be 2 x 2 matrices. If AB = AC, then B...
Let A and B be n × m, and m × n matrices over F respectively. Prove that rn ) = det(In-AB) = det(I,n-BA). In det A
Let A and B be n × m, and m × n matrices over F respectively. Prove that rn ) = det(In-AB) = det(I,n-BA). In det A
(5) Let A, B be two 3 x 3 matrices with eigenvectors v1, 2, vs and w, w2, w3 respectively Under which conditions AB- BA?
9. A square matrix A is said to be nilpotent if A 0 for some integer r 21. Let A, B be nilpotent matrices, of the same size, and assume AB BA. Show that AB and A +B are nilpotent
9. A square matrix A is said to be nilpotent if A 0 for some integer r 21. Let A, B be nilpotent matrices, of the same size, and assume AB BA. Show that AB and A +B are nilpotent
8. Let A and B be two 3x3 matrices such that 3A = A^2 + AB and
detA != 0.
a) determine the expression for B in terms of A
b) show that if A is an eigen value of B with corresponding
eigenvector u then mu = 3 - () is an eigen value of A with
corresponding eigen vector u
c) let B = [2,0,-1;0,2,-1;-1,0,1] find A
8.let Aand B t33 matries snch 3A-AtA and detA70 3 Ca)...
2. (a) Consider the following matrices: A = [ 8 −6, 7 1] , B = [
3 −5, 4 −7] C = [ 3 2 −1 ,−3 3 2, 5 −4 −3 ]
(i) Calculate A + B,
(ii) Calculate AB
(iii) Calculate the inverse of B,
(iv) Calculate the determinant of C.
(b) The points P, Q and R have co-ordinates (2, 2, 1), (4, 1, 2)
and (5, −1, 4) respectively.
(i) Show that P Q~ =...
1) Let A and B be nxn matrices. Show that if I is a nonzero eigenvalue of AB, then it is also an eigenvalue of BA.
8. Let Maxn denote the vector space of all n x n matrices. a. Let S C Max denote the set of symmetric matrices (those satisfying AT = A). Show that S is a subspace of Mx. What is its dimension? b. Let KC Maxn denote the set of skew-symmetric matrices (those satisfying A' = -A). Show that K is a subspace of Max. What is its dimension?
I will rate if correct
4. (10 pts) Let A,B be square matrices with the same size n × n, and let c be a constant. True or False: (a) (AB)-1- B-1A-1 (b) ABメBA in general. (c) det(AB) = det(B) * det(A) (d) (CAB)1A (e) rank(A+ B) S rank(A) + rank(B)