lecture08 - Lecture Lecture 8 2.5 Determinants Chapter 2...

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ecture 8 Lecture 8 5 eterminants 2.5 Determinants Chapter 2 Matrices 1
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= 1 3 4 4 2 3 B Try this … 4 2 0 cofactor expansion along row 1 31 41 43 3 ) (2 ) 4 34 + = 11 11 12 12 13 13 det( ) b B b B b B = ++ B (3 24 04 02 = −− −+= cofactor expansion along row 2 3 2 3 1 3 4 + −= 21 21 22 22 23 23 det( ) b B b B b B = B 0 2 B bB ? + = Compute Lecture 8 Announcement 2 11 21 12 22 13 23 21 11 22 12 23 13 ? + += 0 0
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Announcement ± Solutions for ² Tutorial 2 ² Lab 1 ² PS3 ± No Lab next week Lecture 8 Announcement 3
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ection 2 5 Section 2.5 Determinants Objectives ow do matrix operations affect determinants? • How do matrix operations affect determinants? • What is the relation between invertibility and determinant? • What is the adjoint of a matrix? •Whatis Cramer’s rule ? Chapter 2 Matrices 4
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Matrices with two identical rows (columns) Theorem 2.5.15 & Example 2.5.16 he determinant of a square matrix with o 1. The determinant of a square matrix with two identical rows is zero. he determinant of a square matrix with o 2. The determinant of a square matrix with two identical columns is zero. 9 3 3 1 1 0 0 1 0 4 2 1 2 4 1 2 2 0 0 4 4 2 4 2 1 4 10 1 2 4 det = 0 Chapter 2 Matrices 5 det = 0 det = 0
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Problem 2.5.17 Prove Theorem 2.5.15.1 by mathematical induction. P n : The determinant of an nxn matrix with two identical rows is zero ab b −= x2: Base case 0 bc 2x2: Inductive step (P 2 P 3 ) *** abc 3x3: ** *0 ac = −+ Chapter 2 Matrices 6 cofactor expansion along row 2
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How does e.r.o affect determinants? Discussion 2.5.18 & Theorem 2.5.19 ... ERO B ⎯⎯⎯⎯→ AB What is the relation between det( A ) and det( B )? E.R.O Determinant et( B = k et( i kR det( ) det( ) det( B ) = –det( A ) B A ⎯→ j i R R B A det( B ) = det( A ) B A + j i kR R 222 111 Chapter 2 Matrices 7 135 369 2 = 23 1 3 5 123 = ×
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Using e.r.o. to find determinants Example 2.5.21.1 0 4 2 0 1 2 1 3 1 1 1 3 21 RR 0 4 2 0 0 1 0 0 1 1 1 3 = 1 2 0 0 1 2 0 0 R 31 1 1 2 R 1 1 1 3 23 0240 0010 02 1 43 R2 0 1 0 0 0 4 2 0 = - = 002 6 ) 1 ( 1 2 3 = × × × = 1 0 0 0 Chapter 2 Matrices 8 Gaussian Elimination
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Using e.r.o. to find determinants Example 2.5.21.2 2 1 8 0 5 ⎯→ + 2 9 1 R R 3 2 R R 2 4 R A CD = 9 1 0 0 0 1 2 0 B et( 4 det( 0 3 1 0 0 0 Find det( A ). det( B ) = 4 det( D ) 3 10 3 1 1 ) 2 ( 5 ) det( = × × × = B ) det( ) det( C A = ) det( D = ) det( 4 1 B = 6 5 = Chapter 2 Matrices 9
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To prove: det( B ) = det( A ) Proof of part 3 Theorem 2.5.19 11 12 13 ⎛⎞ aaa 21 Rk R + ⎯⎯⎯⎯→ 22 23 31 32 33 ⎜⎟ = ⎝⎠ A =+ + + B ak a a a the (2, j )-cofactor of A = the (2, j )-cofactor of B A 21 = B 21 A 22 = B 22 A 23 = B 23 Cofactor expansion along row 2 of B : 21 11 21 22 12 22 23 13 23 det( ) (a ka )B ++ B 21 11 21 22 12 22 23 13 23 ka )A 21 21 22 22 23 23 11 21 12 22 13 23 (a A a A a A ) k(a A a A ) + + Chapter 2 Matrices 10 | | det( A ) | | det( A )
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Proof of part 3 To prove: det( B ) = det( A ) Theorem 2.5.19 entries along row 1 want to show = 0
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This note was uploaded on 04/15/2010 for the course MATHS MA1101R taught by Professor Vt during the Spring '10 term at National University of Singapore.

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lecture08 - Lecture Lecture 8 2.5 Determinants Chapter 2...

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