#12. Let A be an m × n matrix and B be an n × p matrix. Prove that(a) N(B) C N(AB).(b) C(AB) C C(A). (Hint: Use Proposition 4.1.)(c) N(B) = N(AB) when A is n × n and nonsingular.(d) C(AB) = C(A) when B is n x n and nonsingular.%3DN --Null spaceC --Column Space

Answered: Image /qna-images/answer/2ed72b5f-6a90-458b-be46-bfafcf67c774.jpg

Answered: #12. Let A be an m × n matrix and B be…

#12. Let A be an m × n matrix and B be an n × p matrix. Prove that (a) N(B) C N(AB). (b) C(AB) C C(A). (Hint: Use Proposition 4.1.) (c) N(B) = N(AB) when A is n x n and nonsingular.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

5. Find the cofactors and the adjoint matrix of A where (1 1 -1 b) A =| 2 3 (7 0 -1 a) A = - 2 0 - 2 4 -3 5 4 5 6 2
1. Indicate whether each statement is true or false. No justification is necessary.
8 - Which of the following statements is FALSE? a) Let A and B be 3 x 3 matrices. If det(A) = 3 and det(B) = 4, then det(AB) = 12. b) Let A be a 4 x 4 matrix. If det(A) = 2 then det(A) =. c) Let A and B be 2 x 2 matrices. Then, AB = BA. d) If det(A) = 1 and det (2A) = 16, then A is a 4 x 4 matrix. e)None
31 32 = 1 + 3x2 1 + 2x3 + 324 Y3 3x2 + x3 Write the matrix A such that y = Ax |
7. Prove that ||A|| = Omax (A) for a real matrix A.
For matrix A = 15 A. 12 c. (15 -203) 20 4). 3 4 if f(x) = x²-2X, what is the answer to f(A)? 15 B. (112 -230) -15 20 D. 12 23 12 23
-12²2 J]. 0 Matrices A and B satisfy AB=B¹, where B = Find (i) without finding B¹, the value of K for which KA-2B¹ + 1 = 0. (ii) without finding A¹, the matrix X satisfying AXA = B.
1) (i) Given any m x n matrix, A, show that A· A¹ and A¹. A must both be defined and T (ii) both must be square matrices. (iii) Show that (A.A¹) = A.A¹ for any matrix A. (Prove these results in general. Do not just give an example.)
2 ... Let A and B be 3 × 3 matrices such that A3 = B| 6 The matrix C = A – 2B is invertible. [5] A) True False
Q1. [2 (a) Given the matrix A = |7 4 -61 show that the matrix (A + A") is -2 4 Symmetrical Matrix. (5 marks) (b) If the matrix A = 3 show that A2 – 4A + 71 = 0. (5 marks) 2.
Q5. Diagonalise the matrix [4]. 3 31 [A] = |3 6 1 [3 4. 1 6 1
70. Consider the matrix M = (ab) where a and c are real and b may be complex. If M can be diagonalized by a matrix S, then S is: