section15 - Chapter 1 MAT188H1F Lec03 Burbulla Chapter 1...

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Unformatted text preview: Chapter 1 MAT188H1F Lec03 Burbulla Chapter 1 Lecture Notes Fall 2007 Chapter 1 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 1 Chapter 1 Matrix Inverses Chapter 1 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 1 Matrix Inverses Example 4 From Section 1.2, Again Consider the system of equations in three variables x , y and z : 2 x + y- z = 1 x + y + z = 6- 3 x + 2 y + 4 z = 13 This can be written as a single matrix equation AX = B , with A = 2 1- 1 1 1 1- 3 2 4 , X = x y z , B = 1 6 13 . Now consider the matrix C = 1 8 - 2 6- 2 7- 5 3- 5 7- 1 . Chapter 1 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 1 Matrix Inverses CA = I We have CA = 1 8 - 2 6- 2 7- 5 3- 5 7- 1 2 1- 1 1 1 1- 3 2 4 = 1 8 8 0 0 0 8 0 0 0 8 = I , as you may check. The matrix C can be used to solve the system of equations in a different way: AX = B C ( AX ) = CB ( CA ) X = CB IX = CB X = CB . That is, the solution to the system is X = 1 8 - 2 6- 2 7- 5 3- 5 7- 1 1 6 13 = 1 8 8 16 24 = 1 2 3 . Chapter 1 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 1 Matrix Inverses The Inverse of a Matrix If A is an n n matrix and C is another n n matrix such that CA = I and AC = I , then C is called the inverse of A , and we write C = A- 1 . Of course, A is also the inverse of C ; you could just as well write A = C- 1 . If a matrix has an inverse, it is said to be invertible. Warning: not all matrices are invertible. Negative powers of an invertible matrix are defined as follows: A- k = ( A- 1 ) k , if k > . Chapter 1 Lecture Notes MAT188H1F Lec03 Burbulla Chapter 1 Matrix Inverses The Gaussian Algorithm for Finding Matrix Inverses How can you find the inverse of a matrix, A ? One way is to use row reduction on the augmented matrix ( A | I ) . Reduce the left side...
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This note was uploaded on 06/22/2008 for the course ENGINEERIN MAT188 taught by Professor Burbula during the Fall '07 term at University of Toronto- Toronto.

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section15 - Chapter 1 MAT188H1F Lec03 Burbulla Chapter 1...

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