oru 2 Let A and B be two n x n matrices. There exists a nonsingular matrix P such that PB = AP. Then which of the following is always true? a) A and B are not similar b) A and B have the same eigenvalues c) A does not have any characteristic polynomial d) B does not have any characteristic polynomial
Problem 5. Let n N. The goal of this problem is to show that if two real n x n matrices are similar over C, then they are also similar over IK (a) Prove that for all X, y є Rnxn, the function f(t) det (X + ty) is a polynomial in t. (b) Prove that if X and Y are real n × n matrices such that X + ừ is an invertible complex matrix, then there exists a...
Let A and B be square matrices of order 3 such that |A| = 4 and |B| = 7 (1) Find |AB|. (2) Find |2A|. (3) Are A and B singular or nonsingular? Explain. (A) A and B are both singular because they both have nonzero determinants. (B) A and B are both nonsingular because they both have nonzero determinants. (C) A is singular, but B is nonsingular because |A| < |B|. (D) B is singular, but A is nonsingular...
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.
Let A be an m x n matrix and let B be an n x p matrix. (a) Prove that Col(AB) SColA) (b) Use part (a) to prove that the rank of AB is at most the rank of A (c) Use transpose matrices to prove that the rank of AB is also at most the rank of B.
Problem 1. Let A be an m x m matrix. (a) Prove by induction that if A is invertible, then for every n N, An is invertible. (b) Prove that if there exists n N such that An is invertible, then A is invertible. (c) Let Ai, . . . , An be m x m matrices. Prove that if the product Ai … An is an invertible matrix, then Ak is invertible for each 1 < k< n. (d)...
Let A, B be an n × n matrices. Prove that [10%] ABt = B tA t
c) Let Ae R"n be nonsingular and let -be any natural matrix norm on R be an eigenvalue of A. Prove that 1/| A-1|| S AS 11A|l. Let A
Two n x n matrices A and B are called similar if there is an invertible matrix P such that B = P-AP. Show that two similar matrices enjoy the following properties. (a) They have the same determinant. (b) They have the same eigenvalues: specifically, show that if v is an eigenvector of A with eigenvalue 1, then P-lv is an eigenvector of B with eigenvalue l. (c) For any polynomial p(x), P(A) = 0 is equivalent to p(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