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lect1116Determinants

# lect1116Determinants - MATH 17100 28446 Determinants The...

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MATH 17100 28446 11/16/2009 Determinants The determinant of a square matrix A is a scalar denoted by det or A A obtained as follows: The determinant of the 1 1 × matrix 11 [ ] a = A is 11 11 det det[ ] a a = A . The determinant of the 2 2 × matrix 11 12 21 22 a a a a = A is 11 12 11 22 12 21 21 22 det det a a a a a a a a = = = A . The determinants of higher order matrices are obtained by any of the following methods: (I) Method of Laplace Expansion Corresponding to the element ij a of the square matrix A , there is a sub-matrix obtained by eliminating the row and column occupied by the element. Let us denote that sub-matrix by ij S . The determinant of this sub-matrix is called the ( i , j ) minor of A and denoted det ij ij M = S . The ( i , j ) cofactor of A is defined as ( 1) i j ij ij A M + = . The determinant of an n n × matrix A is obtained by a cofactor expansion along any row or column of A : 1 det n ij ij j a A = = A , cofactor expansion along the th i row 1 det n ij ij i a A = = A

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