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slide3-aug26 - ECON 401: Determinant and Inverse of a...

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ECON 401: Determinant and Inverse of a Matrice Siyang Xiong Rice University August 25, 2011 Xiong (Rice University) ECON 401 August 25, 2011 1 / 20
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Matrice n equations of n variable a 11 x 1 C a 12 x 2 C ... C a 1 n x n D b 1 (1) a 21 x 1 C a 22 x 2 C ... C a 2 n x n D b 2 ... a n 1 x 1 C a n 2 x 2 C ... C a nn x n D b n (3) can be represented as follows: 2 4 a 11 ... a 1 n ... ... ... a n 1 ... a nn 3 5 2 4 x 1 ... x n 3 5 D 2 4 b 1 ... b n 3 5 Xiong (Rice University) ECON 401 August 25, 2011 2 / 20
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we use j A j to denote the determinant of a n n matrice A for a 1 1 matrice A D [ a ] [ a ] D a . Given a n n matrice A , let A . i , j / denote the matrice resulted by deleting the i -th row and j -th column of A . C ij D . ± 1 / i C j j A . i , j / j is called the (i,j)-th cofactor of A ; then, j A j D a i 1 . ± 1 / i C 1 j A . i , j / jC ... C a in C in D a 1 j C 1 j C ... C a nj C nj Xiong (Rice University) ECON 401 August 25, 2011 3 / 20
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determinant of a matrice: examples ± a b c d D a . ± 1 / 1 C 1 [ d ] C b . ± 1 / 2 C 1 [ c ] D ad ± bc . ± a b c d D b . ± 1 / 1 C 2 [ c ] C d . ± 1 / 2 C 2 [ a ] D ± bc C ad . Xiong (Rice University) ECON 401 August 25, 2011 4 / 20
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2 4 1 1 6 2 4 3 2 4 -5 3 5 D 1 . ± 1 / 1 C 1 ± 4 3 4 -5 C 1 . ± 1 / 1 C 2 ± 2 3 2 -5 C 6 . ± 1 / 1 C 3 ± 2 4 2 4 D [ 4 . ± 5
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This note was uploaded on 02/09/2012 for the course ECON 401 taught by Professor Siyang during the Spring '11 term at Rice.

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slide3-aug26 - ECON 401: Determinant and Inverse of a...

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