04-04-03 - Lecture 24 Andrei Antonenko April 4, 2003 1 On...

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Unformatted text preview: Lecture 24 Andrei Antonenko April 4, 2003 1 On the row(column) expansion This lecture we will give a nice example of application of a row expansion to computing the determinant of a large matrices. Let A n be the matrix with n rows and n columns of the following form. A n = 2 1 0 0 ... 0 0 1 2 1 0 ... 0 0 0 1 2 1 ... 0 0 0 0 1 2 ... 0 0 . . . . . . . . . . . . . . . . . . . . . 0 0 0 0 ... 2 1 0 0 0 0 ... 1 2 So it is a matrix with 2s on a diagonal and 1s strictly above and below it. Our goal is to compute the determinant of A n as a function of n . First we will use expansion by the first row. We will have: det A n = fl fl fl fl fl fl fl fl fl fl fl fl fl fl fl fl fl 2 1 0 0 ... 0 0 1 2 1 0 ... 0 0 0 1 2 1 ... 0 0 0 0 1 2 ... 0 0 . . . . . . . . . . . . . . . . . . . . . 0 0 0 0 ... 2 1 0 0 0 0 ... 1 2 fl fl fl fl fl fl fl fl fl fl fl fl fl fl fl fl fl = 2 fl fl fl fl fl fl fl fl fl fl fl fl fl fl 2 1 0 ... 0 0 1 2 1 ... 0 0 0 1 2 ... 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 ... 2 1 0 0 0 ... 1 2 fl fl fl fl fl fl fl fl fl fl fl fl fl fl- 1 fl fl fl fl fl fl fl fl fl fl fl fl fl fl 1 1 0 ... 0 0 0 2 1 ... 0 0 0 1 2 ... 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 ... 2 1 0 0 0 ... 1 2 fl fl fl fl fl fl fl fl fl fl fl fl fl fl = 2det A n- 1- fl fl fl fl fl fl fl fl fl fl fl fl fl fl 1 1 0 ... 0 0 0 2 1 ... 0 0 0 1 2 ... 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 ... 2 1 0 0 0 ... 1 2 fl fl fl fl fl fl fl fl fl fl fl fl fl fl 1 For the last matrix we will use expansion by the first column. So, the initial determinant will be equal to: det A n = 2det A n- 1- 1 fl fl fl fl fl fl fl fl fl fl fl fl 2 1 ... 0 0 1 2 ... 0 0 . . . . . . . . . . . . . . ....
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04-04-03 - Lecture 24 Andrei Antonenko April 4, 2003 1 On...

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