Chapter 2.3 Determinants-Properties

A is singular implies det a 0 and ab is singular so

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Unformatted text preview: f Proof. A is singular implies det A = 0 and AB is singular, so det(AB ) = 0. Thus, det(AB ) = 0 = 0 · det B = (det A)(det B ). A is invertible implies A = E1 E2 · · · Er for some elementary matrices E1 , E2 , . . . , Er . Hence, det(AB ) = det((E1 E2 · · · Er )B ), so det(AB ) = (det E1 )(det E2 ) · · · (det Er )(det B ). Also, (det E1 )(det E2 ) · · · (det Er ) = det(E1 E2 · · · Er ) = det A Therefore det(AB ) = (det A)(det B ). Outline Linearity of the Determinant Determinants and Invertibility Determinants and Products Determinants and Inverses The Determinant of an Inverse Theorem If A is an invertib...
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This note was uploaded on 02/10/2014 for the course MATH 2270 taught by Professor Kenyonj.platt during the Spring '14 term at Snow College.

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