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Theorem 6: Multiplicative Property for Determinants If A and B are nxn matrices, then det(AB)= (det A)(det B)

Theorem 6: Multiplicative Property for Determinants If A and B are nxn matrices, then det(AB)= (det A)(det B)

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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2. Problem 2: You will be partially proving the following theorem: Theorem 6: Multiplicative Property for Determinants If A and B are nxn matrices, then det(AB) = (det A)(det B) Start of proof: If A is invertible, then A is row equivalent to /. Assuming it takes three elementary row operations to reduce A to the identity matrix, there exists elementary matrices E1, E2, and E, such that E3E,E,A = I. (a) Solve E,E,E,A = I for matrix A. (b) Right multiply both sides of your solution from part (a) by matrix B. (Hint, the answer/result is AB = E'E,'E,'B) (c) Therefore, |AB| = E'E;'E,'B . Use this to carefully show that |AB| = |A||B| Hints: (i) Each E' is an elementary matrix. (ii) You can "expand" or "peel apart" E,'E;'E;'B using methods from problem 1. (iii) You can put "part of that product from (ii) back together". "Carefully" here may require doing this in 2 steps.
Transcribed Image Text:2. Problem 2: You will be partially proving the following theorem: Theorem 6: Multiplicative Property for Determinants If A and B are nxn matrices, then det(AB) = (det A)(det B) Start of proof: If A is invertible, then A is row equivalent to /. Assuming it takes three elementary row operations to reduce A to the identity matrix, there exists elementary matrices E1, E2, and E, such that E3E,E,A = I. (a) Solve E,E,E,A = I for matrix A. (b) Right multiply both sides of your solution from part (a) by matrix B. (Hint, the answer/result is AB = E'E,'E,'B) (c) Therefore, |AB| = E'E;'E,'B . Use this to carefully show that |AB| = |A||B| Hints: (i) Each E' is an elementary matrix. (ii) You can "expand" or "peel apart" E,'E;'E;'B using methods from problem 1. (iii) You can put "part of that product from (ii) back together". "Carefully" here may require doing this in 2 steps.
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