lecture07 - Lecture Lecture 7 2.4 Elementary Matrices 2.5...

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ecture 7 Lecture 7 4 lementary Matrices 2.4 Elementary Matrices 2.5 Determinants Chapter 2 Matrices 1
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Announcement ractice Session 1 and 2 Scores Gradebook ± Practice Session 1 and 2 Scores in Gradebook ± Lab Session 1 this week o lab session ext week ± No lab session next week ± NUS 1-week Term break after next week Lecture 7 Announcement 2
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ection 2 4 Section 2.4 Elementary Matrices Objectives ow to find the inverse of an invertible matrix? • How to find the inverse of an invertible matrix? • How to tell whether a matrix is invertible? • What can we say about an invertible matrix? Chapter 2 Matrices 3
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How to find inverse matrix? Discussion 2.4.6 (A Method to find Inverses) e can use lementary row operations find the We can use elementary row operations to find the inverse of an invertible matrix. very inverse matrix is “made up” of lementary Every inverse matrix is made up of elementary matrices . Chapter 2 Matrices 4
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How to find inverse matrix? Example 2.4.7 Find the inverse of = 0 1 0 0 0 1 I = 3 5 2 3 2 1 A 1 0 0 if it exists. 8 0 1 Form the 3x6 augmented matrix 0 0 1 3 2 1 1 0 0 8 0 1 0 1 0 3 5 2 Chapter 2 Matrices 5 Gauss-Jordan Elimination
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How to find inverse matrix? Example 2.4.7 0 0 1 3 2 1 1 2 2 R R 0 0 1 3 2 1 1 0 0 8 0 1 0 1 0 3 5 2 1 3 R R ⎯→ 1 0 1 5 2 0 0 1 2 3 1 0 + 2 3 2 R R 0 1 2 3 1 0 0 0 1 3 2 1 ⎯→ 3 R 0 1 2 3 1 0 0 0 1 3 2 1 1 2 5 1 0 0 1 2 5 1 0 0 A –1 I 3 1 3 R R ⎯→ 1 2 5 1 0 0 3 5 13 0 1 0 3 6 14 0 2 1 ⎯→ 2 1 2 R R 1 2 5 1 0 0 3 5 13 0 1 0 9 16 40 0 0 1 Chapter 2 Matrices 6 3 2 3 R R + RREF
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Why does it work? Question: Given any invertible matrix , is the RREF always I ? Discussion 2.4.6 A : invertible matrix of order n Suppose the RREF of A is the identity matrix E 1 E 2 E k I A ⎯→ k R R R " 2 1 elementary matrices orm an “ ugmented matrix” ( E k ··· E 2 E 1 A = I E k ··· E 2 E 1 = A –1 Not practical Form an n x 2 n augmented matrix A | I ) (| ) " 21 k EE E A I applying e.r.o. to ( A | I ) same as ) | ( k k I E E E A E E E 1 2 1 2 " " = ) | ( 1 = A I applying e.r.o. to both A and I Chapter 2 Matrices 7 ( A | I ) ö ( I | A –1 ) Gauss-Jordan Elimination
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A very 3 important theorem Theorem 2.4.5 Let A be a square matrix. Any one of the statements implies the other three. 1. A is invertible . The following statements are equivalent . 2. The linear system Ax = 0 has only the trivial solution . 3. The reduced row-echelon form of A is an identity matrix .
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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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lecture07 - Lecture Lecture 7 2.4 Elementary Matrices 2.5...

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