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Today s goal The inverse of a matrix Composition rules An algorithm for computing the inverse matrix Math 415 Lecture 6 Matrix inverses February 1th 2016 Today s goal The inverse of a matrix Textbook Chapter 1 6 Composition rules An algorithm for computing the inverse matrix Today s goal The inverse of a matrix Composition rules An algorithm for computing the inverse matrix Textbook Chapter 1 6 Suggested Practice Exercise Chapter 1 6 Exercise 1 2 4 6 10 11 18 35 36 37 38 40 49 50 Today s goal The inverse of a matrix Composition rules An algorithm for computing the inverse matrix Textbook Chapter 1 6 Suggested Practice Exercise Chapter 1 6 Exercise 1 2 4 6 10 11 18 35 36 37 38 40 49 50 Khan Academy Video Inverse Matrix part I Inverse Matrix part II Today s goal The inverse of a matrix Composition rules Today s goal An algorithm for computing the inverse matrix Today s goal The inverse of a matrix Composition rules An algorithm for computing the inverse matrix Elementary matrices perform row operations 1 0 0 a b c a b c 2 1 0 d e f 2a d 2b e 2c f 0 0 1 g h i g h i Today s goal The inverse of a matrix Composition rules An algorithm for computing the inverse matrix Elementary matrices perform row operations 1 0 0 a b c a b c 2 1 0 d e f 2a d 2b e 2c f 0 0 1 g h i g h i We know how to reverse a single row operation 1 1 0 0 1 0 0 a 1 0 a 1 0 0 0 1 0 0 1 Today s goal The inverse of a matrix Composition rules An algorithm for computing the inverse matrix Elementary matrices perform row operations 1 0 0 a b c a b c 2 1 0 d e f 2a d 2b e 2c f 0 0 1 g h i g h i We know how to reverse a single row operation 1 1 0 0 1 0 0 a 1 0 a 1 0 0 0 1 0 0 1 Inverting a more complicated matrix is harder 1 1 0 0 1 0 0 a 1 0 a 1 0 0 b 1 ab b 1 Goal today how to find an inverse to any square matrix Today s goal The inverse of a matrix Composition rules An algorithm for computing the inverse matrix Elementary matrices perform row operations 1 0 0 a b c a b c 2 1 0 d e f 2a d 2b e 2c f 0 0 1 g h i g h i We know how to reverse a single row operation 1 1 0 0 1 0 0 a 1 0 a 1 0 0 0 1 0 0 1 Inverting a more complicated matrix is harder 1 1 0 0 1 0 0 a 1 0 a 1 0 0 b 1 ab b 1 Goal today how to find an inverse to any square matrix Today s goal The inverse of a matrix Composition rules An algorithm for computing the inverse matrix Elementary matrices perform row operations 1 0 0 a b c a b c 2 1 0 d e f 2a d 2b e 2c f 0 0 1 g h i g h i We know how to reverse a single row operation 1 1 0 0 1 0 0 a 1 0 a 1 0 0 0 1 0 0 1 Inverting a more complicated matrix is harder 1 1 0 0 1 0 0 a 1 0 a 1 0 0 b 1 ab b 1 Goal today how to find an inverse to any square matrix Today A will be an n n matrix Today s goal The inverse of a matrix Composition rules An algorithm for computing the inverse matrix The inverse of a matrix Today s goal The inverse of a matrix Composition rules An algorithm for computing the inverse matrix The inverse of a real number a is denoted by a 1 For example 7 1 1 7 and 7 7 1 7 1 7 1 Today s goal The inverse of a matrix Composition rules An algorithm for computing the inverse matrix The inverse of a real number a is denoted by a 1 For example 7 1 1 7 and 7 7 1 7 1 7 1 Remember that the identity matrix 1 0 0 1 In In is the n n matrix 0 0 0 0 1 Today s goal The inverse of a matrix Composition rules An algorithm for computing the inverse matrix The inverse of a real number a is denoted by a 1 For example 7 1 1 7 and 7 7 1 7 1 7 1 Remember that the identity matrix 1 0 0 1 In In is the n n matrix 0 0 0 0 1 Definition An n n matrix A is said to be invertible if there is an n n matrix C satisfying CA AC In Today s goal The inverse of a matrix Composition rules An algorithm for computing the inverse matrix The inverse of a real number a is denoted by a 1 For example 7 1 1 7 and 7 7 1 7 1 7 1 Remember that the identity matrix 1 0 0 1 In In is the n n matrix 0 0 0 0 1 Definition An n n matrix A is said to be invertible if there is an n n matrix C satisfying CA AC In Today s goal The inverse of a matrix Composition rules An algorithm for computing the inverse matrix The inverse of a real number a is denoted by a 1 For example 7 1 1 7 and 7 7 1 7 1 7 1 Remember that the identity matrix 1 0 0 1 In In is the n n matrix 0 0 0 0 1 Definition An n n matrix A is said to be invertible if there is an n n matrix C satisfying CA AC In where In is the n n identity matrix We call C the Today s goal The inverse of a matrix Composition rules An algorithm for computing the inverse matrix The inverse of a real number a is denoted by a 1 For example 7 1 1 7 and 7 7 1 7 1 7 1 Remember that the identity matrix 1 0 0 1 In In is the n n matrix 0 0 0 0 1 Definition An n n matrix A is said to be invertible if there is an n n matrix C satisfying CA AC In where In is the n n identity matrix We call C the inverse of A Today s goal The inverse of a matrix Composition rules An algorithm for computing the inverse matrix Example We already know that an elementary matrix is invertible 1 1 0 0 1 0 0 8 1 0 8 1 0 0 0 1 …
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