TUE7Quiz3 - A and B are 2 2 matrices, then det( M ) = det(...

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Linear Algebra Class Tue 7 Quiz 3 ID: Name: 1 Indicate whether the statement is true( T ) or false( F ). Justify your answer. (a) If A T = A - 1 , then det( A ) = ± 1. ( T ) { det( A ) } 2 = det( A )det( A T ) = det( A )det( A - 1 ) = det( I ) = 1 (b) If a square matrix A can be reduced to row echelon form by Gaussian elimination with no row interchanges, then A has an LU-decomposition that is unique. ( F ) 1 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 = 1 0 0 0 1 0 0 2 1 1 0 0 0 0 0 0 0 0 2 Find an LU-decomposition of A A = 1 1 0 0 0 2 1 2 0 0 2 0 0 1 2 2 0 0 0 0 0 4 0 0 0 0 0 0 1 2 0 0 0 4 1 0 LU = 1 0 0 0 0 0 1 1 0 0 0 0 0 1 2 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 4 1 - 2 1 1 0 0 0 2 0 1 0 0 2 - 2 0 0 1 1 - 1 1 0 0 0 1 0 0 0 0 0 0 1 2 0 0 0 0 0 1 3 Prove that if a square matrix M is partitioned into block triangular form as M = A C 0 B ! in which
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Unformatted text preview: A and B are 2 2 matrices, then det( M ) = det( A )det( B ). m ij = 0 for i = 1 , 2 , j = 3 , 4 det( M ) = X m 1 j 1 m 2 j 2 m 3 j 3 m 4 j 4 = a 11 a 22 b 11 b 22-a 11 a 22 b 12 b 21-a 12 a 21 b 11 b 22 + a 12 a 21 b 12 b 21 = a 11 a 22 ( b 11 b 22-b 12 b 21 )-a 12 a 21 ( b 11 b 22-b 12 b 21 ) = ( a 11 a 22-a 12 a 21 )( b 11 b 22-b 12 b 21 ) = det( A )det( B ) Oct. 9, 2007 Typeset by L A T E X...
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This note was uploaded on 05/19/2008 for the course MS 4032 taught by Professor Anony during the Spring '08 term at A.T. Still University.

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