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ALA240-2009 - November 26th

# ALA240-2009 - November 26th - November 26 th 2009 MAT240...

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Unformatted text preview: November 26 th , 2009 MAT240: Abstract Linear Algebra Lecture: Determinants: 1. Applications: Mention two, prove one, use none. 2. Formulas: discuss just one. 3. Basic Properties. Our core subject. Det: ?¡? ( ? ) → ? Use 1: ??? − − −¢ 1 − − − ⋮ − − −¢ ? − − − £ = ¤¥¦§¨¢¨¥¥?¥¤§©§?? ª§¨???? «¬ ¢ 1 … ¢ ? ­ Use 2: det ¦®­ ≠ 0 ©?? ® − 1 ∃ Definition: ¯¦¨ 11 ­¯ ∶ = ¨ 11 ¯®¯ = °± ¨ 11 ⋯ ¨ 1 ? ⋮ ⋱ ⋮ ¨ ? 1 ⋯ ¨ ?? ²° ≡ ³ ( − 1) © + ´ ¨ ©´ µ® ©´ ¶ µ ? ´ =1 in general, ® ©´ ¶ is the matrix obtained from A by removing the © ?ℎ row and ´ ?ℎ column. Eg. ·¸ ¨ ¹ « ? º· = ¨? − «¹ Eg. ·¸ ¨ 11 ¨ 12 ¨ 21 ¨ 22 º· = ¦− 1 ­ 1+1 ∗ ¨ 11 ¯¦¨ 22 ­¯ = ¨ 11 ¨ 22 − ¨ 12 ¨ 21 Eg. ·¸ 1 2 3 4 º· = 1 ∗ 4 − 2 ∗ 3 = − 2 Eg. ° 1 2 3 4 5 6 7 8 9 £° = 1 ·¸ 5 6 8 9 º· − 2 ·¸ 4 6 7 9 º· + 3 ·¸ 4 5 7 8 º· = 1 ¦− 3 ­ − 2 ¦−...
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