Are the following statements true or false? Justify with a proof (if true) or acounterexample (if false).(a) For any (m X n) matrix A and any (n X m) matrix B, if AB = Im,thenBA = In.(b) If A and B are two (n X n) matrices such that BA is invertible, then A isinvertible.

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Answered: Are the fOllowing statem ue of faise?…

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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6. Prove a shortened version of the fact from the prior homework using theory of Ch. 3 [if you are utilizing theorems or prior results, cite them!]. Let A and B be square matrices. If AB = I, then A and B are both invertible.
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
Consider the n ×n matrices A and B and their product AB(a) Show that if AB is invertible, then so is A.(Hint: There is a matrix W such that ABW = I. Why?) (b) Show that if AB is invertible, then so is B.
a) Let A= 04 1 0 Find a diagonal matrix D so find matrix P) that PAP=D for some invertible matrix P. (Don't need to
3) We know, by theorem, that if A is a square matrix then the following statements are equivalent: (a) A is invertible. (b) AX=0 has only the trivial solution. (c) The RREF of A is I. (d) A is expressible as a product of elementary matrices. 24 Consider the following matrix, A = [34] 01 Verify that all 4 conditions in the theorem are either all true or all false for the given matrix. That is, if A is invertible, show that the 3 other conditions are true, as well. If it is not, show that they are all false.
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.)
10. Prove that every 3 x 3 matrix is a linear combination of invertible matrices. Proof:
True or false
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
If A, B are two n × n orthogonal matrices, then AB is also an orthogonal matrix. True False

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Are the fOllowing statem ue of faise? Justify with a e) or a counterexample (if false). (a) For any (m × n) matrix A and any (n X m) matrix B, if AB = Im, then ВА In. (b) If A and B are two (n × n) matrices such that BA is invertible, then A is invertible.