c4s3 - 4.3 Computation of Determinants and Cramers Rule 1...

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4.3 Computation of Determinants and Cramer’s Rule 1 Chapter 4. Determinants 4.3 Computation of Determinants and Cramer’s Rule Note. Computation of A Determinant. The determinant of an n × n matrix A can be computed as follows: 1. Reduce A to an echelon form using only row (column) addition and row (column) interchanges. 2. If any matrices appearing in the reduction contain a row (column) of zeros, then det( A )=0 . 3. Otherwise, det( A )=( 1) r · (product of pivots) where r is the number of row (column) interchanges. Example. Page 271 number 6 (work as in Example 1 of page 264).

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4.3 Computation of Determinants and Cramer’s Rule 2 Theorem 4.5. Cramer’s Rule. Consider the linear system A~x = ~ b ,where A =[ a ij ]isan n × n invertible matrix, ~x = x 1 x 2 . . . x n and ~ b = b 1 b 2 . . . b n . The system has a unique solution given by x k = det( B k ) det( a ) for k =1 , 2 ,...,n, where B k is the matrix obtained from A
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This note was uploaded on 02/24/2012 for the course MATH 285 taught by Professor Igordolgachev during the Fall '04 term at University of Michigan-Dearborn.

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c4s3 - 4.3 Computation of Determinants and Cramers Rule 1...

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