I am using Textbook: Linear Algebra: A Modern Introduction by David Poole, Cengage Learning ISBN-13: 978-1285463247
Chapter 3: Matrices
Chapter 4: Eigenvalues and Eigenvectors
5. Solve the following matrix equations for X. Simplify your answer as much as possible. (a) (A 1 X) 1 = A(B 2 A) 1 . (b) ABXA 1 B 1 = I A. 6. (a) Prove that I Ak = (I A)(I A A2 . . . Ak 1 ) for any square matrix A and k = 1, 2, . . .. (b) Prove that if Ak = O then I A is invertible and (I A) 1 = 2 k 1 I A A … A . 7. (a) Prove by induction that (A1 A2 . . . Ak )T = AT1 AT2 . . . ATk for any m ⇥ n matrices A1 , A2 , . . . , Ak . (b) Prove by induction that (A1 A2 . . . Ak ) 1 = Ak 1 Ak 1 1 . . . A1 1 for any n ⇥ n invertible matrices A1 , A2 , . . . , Ak . 8. For each square matrix C denote by Tr(C) the P sum of all entries on its n diagonal. That is, if C = (cij )n⇥n then Tr(C) = i=1 cii . Prove that (a) Tr(A B) = Tr(A) Tr(B) for any n ⇥ n matrices A and B. (b) Tr(AB) = Tr(BA) for any m ⇥ n matrix A and n ⇥ m matrix B. 9. Given the matrix 2 1 A = 40 1 1 2 1 3 3 1 5. 4 (a) Give a base for row(A), col(A), and null(A). (b) Find rank(A), and nullity(A). 10. Let (2 3 2 3) 4 6 A = 4 35 , 4 35 , 2 1 ( 243 263 213 ) 435 , 435 , 415 , 2 1 1 ( 243 263 213 243 ) C = 435 , 435 , 415 , 425 . 2 1 1 1 B= (a) Which of above set of vectors are linearly independent? Explain. (b) Which of above set of vectors span R3 ? Explain. (c) Which of above set of vectors is a basis of R3 ? Explain. 2