Provide complete arguments/proofs for the following. a) A is invertible iff there exists a matrix B so that AB = BA = I. It is simple to show that: (i) If A is invertible and AB = I, then BA = I as well and B is the unique such matrix. (ii) If A is invertible and BA = I, then AB = I as well and B is the unique such matrix. This shows that if A is invertible, then there is a unique matrix B such that AB = I or BA = I. Call this unique matrix A-1. The goal here is to show that the assumption “A is invertible” is not needed in (i) or (ii). Prove: Let A and B be square matrices with AB = I. Show that A is invertible and hence B = A-1. b) Prove: NS(A) = NS(AT A) for any matrix A. c) Prove: If A and B are m x n matrices such that Ax = Bx for all x∈ Rn, then A = B.
Provide complete arguments/proofs for the following. a) A is invertible iff there exists a matrix B so that AB = BA = I. It is simple to show that: (i) If A is invertible and AB = I, then BA = I as well and B is the unique such matrix. (ii) If A is invertible and BA = I, then AB = I as well and B is the unique such matrix. This shows that if A is invertible, then there is a unique matrix B such that AB = I or BA = I. Call this unique matrix A-1. The goal here is to show that the assumption “A is invertible” is not needed in (i) or (ii). Prove: Let A and B be square matrices with AB = I. Show that A is invertible and hence B = A-1. b) Prove: NS(A) = NS(AT A) for any matrix A. c) Prove: If A and B are m x n matrices such that Ax = Bx for all x∈ Rn, then A = B.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Provide complete arguments/proofs for the following.
- a) A is invertible iff there exists a matrix B so that AB = BA = I. It is simple to
show that:
(i) If A is invertible and AB = I, then BA = I as well and B is the unique such
matrix.
(ii) If A is invertible and BA = I, then AB = I as well and B is the unique such
matrix.
This shows that if A is invertible, then there is a unique matrix B such that AB = I
or BA = I. Call this unique matrix A-1.
The goal here is to show that the assumption “A is invertible” is not needed in (i)
or (ii).
Prove: Let A and B be square matrices with AB = I. Show that A is invertible
and hence B = A-1.
- b) Prove: NS(A) = NS(AT A) for any matrix A.
- c) Prove: If A and B are m x n matrices such that Ax = Bx for all x∈ Rn, then
A = B.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
