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O If E is an elementary matrix then so is E". O Let A and B be m x n matrices and D be an n x n matrix. If D is invertible and (A – B)D = 0 then A= B. %3D

O If E is an elementary matrix then so is E". O Let A and B be m x n matrices and D be an n x n matrix. If D is invertible and (A – B)D = 0 then A= B. %3D

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please solve (d) and (e) only
For each of the following statements, either prove it or give a counterexample to the statement. (a) If A and D are both invertible n x n matrices then so is ADA". (b) If A? = A and A # 0 then A is invertible. (c) If A is an invertible n x n matrix, then the number of solutions of Ax = b depends on the vector b € R". (d) If E is an elementary matrix then so is E". (e) Let A and B be m x n matrices and D be an n x n matrix. If D is invertible and (A – B)D = 0 then A = B.
Transcribed Image Text:For each of the following statements, either prove it or give a counterexample to the statement. (a) If A and D are both invertible n x n matrices then so is ADA". (b) If A? = A and A # 0 then A is invertible. (c) If A is an invertible n x n matrix, then the number of solutions of Ax = b depends on the vector b € R". (d) If E is an elementary matrix then so is E". (e) Let A and B be m x n matrices and D be an n x n matrix. If D is invertible and (A – B)D = 0 then A = B.
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