# lecture06 - Lecture Lecture 6 2.3 Inverses of Square...

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ecture 6 Lecture 6 3 verses of Square Matrices 2.3 Inverses of Square Matrices 2.4 Elementary Matrices Chapter 2 Matrices 1

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Announcement ractice ession 1 Score ± Practice Session 1 Score ² Check it in IVLE Gradebook ab 1 ext week print your worksheet) ± Lab 1 (next week, print your worksheet) ± Solutions of tutorial 1 and exercise set 1 vailable in orkbin (available in workbin) ± Math Clinic Lecture 6 Announcement 2
ection 2 3 Section 2.3 Inverses of Square Matrices Objectives •Whatisan invertible matrix? • What is the inverse of a matrix? • What are some basic properties of invertible atrices? matrices? • What are the powers of a matrix? ther terminologies Chapter 2 Matrices 3 Other terminologies singular / non-singular matrix

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Motivation Discussion 2.3.1 a , b real numbers such that a 0 To solve the equation ax = b = = ( -1 · inverse of a x b / a ( a ) b Let A , B be two matrices. To solve the matrix equation AX = B Can we do this: X = B/A ? Can we find “ inverses ” for matrices We do not have “ division ” for matrices. Chapter 2 Matrices 4 which have the similar property as a -1 ?
For ordinary numbers: (a - = 1 - a = 1 What is an invertible matrix? Definition 2.3.2 A : square matrix of order n . a(a 1 ) = 1 (a 1 )a = 1 A is invertible if there exists a square matrix B of order n A 8 such that AB = I and BA = I The matrix B here is called an inverse of A . oes every matrix have an inverse? o A square matrix is called singular if it has no inverse. Does every matrix have an inverse? No Chapter 2 Matrices 5 non-singular = invertible

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What is an invertible matrix? Example 2.3.3.1 = = 2 1 5 3 3 1 5 2 B A I AB = = = 1 0 0 1 2 1 5 3 3 1 5 2 I BA = = = 1 0 0 1 3 1 5 2 2 1 5 3 So A is invertible and B is an inverse of A lso vertible nd an verse f Chapter 2 Matrices 6 Also B is invertible and A is an inverse of B
x1 variable column matrix matrix equation AX = B A simple application Example 2.3.3.2 2x1 variable column matrix = 0 4 3 1 5 2 X Linear system = 0 4 2 1 5 3 3 1 5 2 2 1 5 3 X = 4 12 1 0 0 1 X = 4 12 X Solution of the linear system Chapter 2 Matrices 7 Given a matrix A , how to find the inverse?

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An example of a singular matrix Example 2.3.3.3 0 1 Show that is singular. No inverse 0 1 A g Proof by Contradiction proving technique Suppose A has an inverse: assume the opposite of the conclusion = d c b a B Let be the inverse write down the object By definition of inverses, = = 1 0 0 1 I BA n the other hand using definition On the other hand, The two results for BA contradict with each other.
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lecture06 - Lecture Lecture 6 2.3 Inverses of Square...

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