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DETAILS POOLELINALG4 3.3.047. Prove that if A and B are square matrices and AB is invertible, then both A and B are invertible. Let A and B be square matrices and assume that AB is invertible. Then, AB(AB)-1 = -Select--- v = I so that A is invertible with A-1 = --Select--- v This means B((AB)-'A) = -.-Select--- = I, and B is invertible, with inverse B-1 = | ---Select-. v. Therefore, both A and B are invertible.

DETAILS POOLELINALG4 3.3.047. Prove that if A and B are square matrices and AB is invertible, then both A and B are invertible. Let A and B be square matrices and assume that AB is invertible. Then, AB(AB)-1 = -Select--- v = I so that A is invertible with A-1 = --Select--- v This means B((AB)-'A) = -.-Select--- = I, and B is invertible, with inverse B-1 = | ---Select-. v. Therefore, both A and B are invertible.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section: Chapter Questions
Problem 17RQ
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DETAILS POOLELINALG4 3.3.047. MY NOTES Prove that if A and B are square matrices and AB is invertible, then both A and B are invertible. Let A and B be square matrices and assume that AB is invertible. Then, AB(AB)-1 = ---Select--- v = I so that A is invertible with A-1 =---Select--- v This means B((AB)-'A) = ---Select--- v = I, and B is invertible, with inverse B-1 = ---Select-.- v Therefore, both A and B are invertible.
Transcribed Image Text:DETAILS POOLELINALG4 3.3.047. MY NOTES Prove that if A and B are square matrices and AB is invertible, then both A and B are invertible. Let A and B be square matrices and assume that AB is invertible. Then, AB(AB)-1 = ---Select--- v = I so that A is invertible with A-1 =---Select--- v This means B((AB)-'A) = ---Select--- v = I, and B is invertible, with inverse B-1 = ---Select-.- v Therefore, both A and B are invertible.
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