lecture9 - Differential Equations & Linear...

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Unformatted text preview: Differential Equations & Linear Algebra Lecture 9 Jianli XIE Email: xjl@sjtu.edu.cn Department of Mathematics, SJTU Fall 2010 Contents Determinants and Cramer’s rule Contents Determinants and Cramer’s rule General theory of n th order linear equations Existence and uniqueness Structure of solutions of homogeneous equations Structure of solutions of nonhomogeneous equations Notations and terminologies A determinant is an expression that is always associated with a square matrix. The notation for the n th order determinant of an n × n matrix A is det A = det( a ij ) = a 11 a 12 · · · a 1 n a 21 a 22 · · · a 2 n . . . . . . · · · . . . a n 1 a n 2 · · · a nn . Minor of a ij , denoted by M ij , is the ( n- 1) th order determinant that results from eliminating row i and column j in det A . Cofactor of a ij , denoted by C ij , is ± M ij , where the sign can be calculated as (- 1) i + j . Cofactor expansion Example If the 2 × 2 matrix A = a b c d , then det A = a b c d = ad- bc . An n th order determinant can be evaluated by cofactor expansion : det A = sum of a ij C ij along any row or column. For example, along row 1 , det A = n X j =1 a 1 j C 1 j = a 11 M 11- a 12 M 12 + a 13 M 13-· · · along column 2 , det A = n X i =1 a i 2 C i 2 =- a 12 M 12 + a 22 M 22- a 32 M 32 + · · · . Example Example By cofactor expansion, the third order determinant det A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 = a 11 · a 22 a 23 a 32 a 33- a 12 · a 21 a 23 a 31 a 33 + a 13 · a 21 a 22 a 31 a 32 = a 11 a 22 a 33- a 11 a 23 a 32- a 12 a 21 a 33 + a 12 a 31 a 23 + a 13 a 21 a 32- a 13 a 31 a 22 Properties of a determinant Theorem a. If one row or column are all zeros, then the determinant is zero. b. The determinant of a triangular matrix is the product of all entries on main diagonal. c. Interchanging two rows or two columns, the determinant reverses sign. d. Multiplying a row or column by a constant k , the entire determinant is multiplied by k . e. Adding a multiple of one row to another, the determinant does not change. f. If two rows or columns are proportional, then the determinant is zero. Example Example Let A n be an n × n matrix of the form A n = 2 1 1 2 1 1 . . . . . . . . . . . . 1 1 2 That is, it is a matrix with 2 ’s on the main diagonal and 1 ’s strictly above and below it. Compute the determinant det A n as a function of n . Solution Expanding the determinant along the first row, we have Example det A n = 2 det A n- 1- 1 1 0 2 1 1 2 ....
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lecture9 - Differential Equations & Linear...

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