ALA240-2009 - December 1st

ALA240-2009 - December 1st - Page 1 of 2 December 1 st 2009...

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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 ª« ¬?­­?®?¯ ­?°°± ²³³³³³³³³³³³³³³´ ¥ 1 … ¥ ? & § are invertible. ↔ det ¥ 1 ¦ ≠ , ±?? det ¥ 2 ¦ ≠ 0 ±?? … ±?? det ⁡ ( § ) ≠ ↔ µ¥ 1 µµ¥ 2 µ … µ§µ ↔ det ¤¦ = det ⁡ ( ¥ 1 … ¥ ?...
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This note was uploaded on 05/25/2011 for the course MAT 244 taught by Professor Christinasaleh during the Fall '10 term at University of Toronto.

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ALA240-2009 - December 1st - Page 1 of 2 December 1 st 2009...

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