{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

This preview shows pages 1–11. Sign up to view the full content.

Lecture 8 2 5 Determinants 2.5 Chapter 2 Matrices 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
= 1 3 4 4 2 3 B Try this … 4 2 0 cofactor expansion along row 1 3 1 4 1 4 3 ( 3) ( 2) 4 34 = + = 11 11 12 12 13 13 det( ) b B b B b B = + + B 2 4 0 4 0 2 − − cofactor expansion along row 2 2 4 3 4 3 2 4 3 1 34 = − + = 21 21 22 22 23 23 det( ) b B b B b B = + + B 2 4 0 4 0 2 11 21 12 22 13 23 b B b B b B ? + + = 0 Compute Lecture 8 Announcement 2 21 11 22 12 23 13 b B b B b B + + = 0
Announcement Solutions for Tutorial 2 Lab 1 PS3 No Lab next week Lecture 8 Announcement 3

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Section 2 5 Section 2.5 Determinants Objectives • How do matrix operations affect determinants? • What is the relation between invertibility and determinant? What is the adjoint of a mat i ? • What is the of a matrix? • What is Cramer’s rule ? Chapter 2 Matrices 4
Theo em 2 5 15 & E ample 2 5 16 Matrices with two identical rows (columns) Theorem 2.5.15 & Example 2.5.16 1 The determinant of a square matrix with two 1. identical rows is zero. 2 The determinant of a square matrix with two 2. identical columns is zero. 9 3 3 1 1 0 0 1 4 10 1 4 2 1 2 4 1 2 2 0 0 4 4 2 4 2 1 2 4 det = 0 Chapter 2 Matrices 5 det = 0 det = 0

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 0 a b ab ab = = 2x2: Base case a b a b c Inductive step (P 2 P 3 ) * * * a b c 3x3: * * * 0 b c a c a b b c a c a b = − + = Chapter 2 Matrices 6 cofactor expansion along row 2
How does e.r.o affect determinants? Discussion 2.5.18 & Theorem 2.5.19 . . . E R O A B ⎯⎯⎯⎯→ What is the relation between det( A ) and det( B )? E.R.O Determinant det( B ) = k det( A ) B A i kR det( ) det( ) det( B ) = –det( A ) kR R B A j i R R det( B ) = det( A ) B A + j i 2 2 2 1 3 5 1 1 1 1 1 1 2 3 1 3 5 Chapter 2 Matrices 7 3 6 9 2 1 3 5 3 6 9 = 1 2 3 = ×

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Using e.r.o. to find determinants Example 2.5.21.1 1 1 1 3 1 1 1 3 0 4 2 0 1 2 1 3 2 1 R R 0 4 2 0 0 1 0 0 = 1 2 0 0 1 2 0 0 2 3 R R 3 1 1 1 R 2R 1 1 1 3 0 2 4 0 0 0 1 0 0 0 2 1 4 3 1 0 0 0 0 1 0 0 0 4 2 0 = - = 6 ) 1 ( 1 2 3 = × × × = Chapter 2 Matrices 8 Gaussian Elimination
Using e.r.o. to find determinants Example 2.5.21.2 + 2 R R R R 4 R 1 8 0 5 ⎯→ 2 9 1 ⎯→ 3 2 ⎯→ 2 A C D = 9 1 0 0 0 1 2 0 B det( B ) 4 det( D ) 10 1 3 1 0 0 0 Find det( A ). ) = 4 det( 3 3 1 ) 2 ( 5 ) det( = × × × = B ) det( ) det( C A = ) det( D = ) det( 4 1 B = 6 5 = Chapter 2 Matrices 9

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
To prove: det( B ) = det( A ) Proof of part 3 Theorem 2.5.19 11 12 13 a a a 11 12 13 a a a 2 1 R kR + ⎯⎯⎯⎯→ 21 22 23 31 32 33 = A a a a a a a 21 11 22 12 23 13 31 32 33 = + + + B a ka a ka a ka 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 (a ka )B (a ka )B = + + + + + B 21 11 21 22 12 22 23 13 23 (a ka )A (a ka )A (a ka )A = + + + + + 21 21 22 22 23 23 11 21 12 22
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 32

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

This preview shows document pages 1 - 11. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online