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Unformatted text preview: Page 1 of 2 December 1 st , 2009 MAT240: Abstract Linear Algebra Lecture: Theorem: If ?? ` = ?? satisfies 0-3, then ?? ` = ?? Theorem: If = 1 ? is a product of elementary matrices and B is a matrix in RREF, then det = det 1 det 2 det ? Proof: det = det 1 2 ? = det det ( `) det ( ) = det 1 det 2 ? = det 1 det 2 det ? det ( ) Theorem: A is invertible det det is invertible. Proof: Write = 1 ? where B is in RREF A is invertible ??? ? 1 ? & are invertible. det 1 , ?? det 2 0 ?? ?? det ( ) 1 2 det = det ( 1 ?...
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