Question 18: If Aand Bare symmetric n xn matrices prove that AB is symmetric or give...
QUESTION 6 a) Prove the product of 2 2 x 2 symmetric matrices A and B is a symmetric matrix if and only if AB=BA. b) Prove the product of 2 nx n symmetric matrices A and B is a symmetric matrix if and only if AB=BA.
Please answer all if possible. Question 15: Do the vectors below form a basis for R3? If so, explain. If not remove as many vectors as you need to form a basis and show that the resulting set of vectors form a basis for R3. C1 = -- () -- () -- () -- (1) Question 16: Carefully consider if equalities below valid for matrices. For each equality state if there is a restriction on dimensions under which they are...
(9) True of False: For all (n xn) matrices A and B, dim(ker(AB)) > dim(ker(B)). (That is, the dimension of the nullspace or kernel of AB is at least as big as the dimension of the nullspace or kernel of B.) Justify your answer. (10) (Extra Credit) Let A be any nxn matrix. If n is odd, prove that it is impossible for im(A) = (A).
19. Suppose A and B are n xn matrices. a. Suppose that both A and B are diagonalizable and that they have the same eigen- vectors. Prove that AB = BA. b. Suppose A has n distinct eigenvalues and AB = BA. Prove that every eigen vector of A is also an eigen vector of B. Conclude that B is diagonalizable. (Query: Need every eigenvector of B be an eigenvector of A?)
7. (10) Find all n xn orthogonal, symmetric, and positive definite real matrix (matrices). Explain your answer.
Let A = CD where C, D are n xn matrices, and is invertible. Prove that DC is similar to A. Hint: Use Theorem 6.13, and understand that you can choose P and P-inverse. Prove that if A is diagonalizable with n real eigenvalues 11, 12,..., An, then det(A) = 11. Ay n Prove that if A is an orthogonal matrix, then so are A and 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 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
Linear Algebra question: If A, B are square matrices and AB is invertible (Inverse), prove that A and B are invertible (Inverse).
vi) Suppose that A and B are two n × n matrices and that AB-A is invertible. Prove that BA-A is also invertible.
4. (5 points) Let A and B ben x n matrices. Prove that if A and B are skew symmetric, then A - B is skew symmetric. Recall C = [cj] is skew symmetric iff Cij =-Cji.