26_hwc1SolnsODDA

26_hwc1SolnsODDA - A 2 = -1 8-4 7 . Therefore, A 2 + 2 A +...

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and so tr( BC ) = n X i,j =1 B i,j C j,i . ( ** ) In the same way, we see that tr( CB ) = n X i,j =1 C i,j B j,i . ( * * * ) Since the names of the variables that we are summing over – the “dummy variables” – do not matter, we can exchange i and j in ( ** ), with the result that tr( BC ) = n X i,j =1 B j,i C i,j . Since B j,i C i,j = C i,j B j,i , we see by comparing the last displayed equation and ( *** ) that tr( BC ) = tr( BC ). 3.29 Consider the matrices A = h 1 2 - 1 3 i and B = ± 1 - 1 0 1 2 1 0 1 3 ² . Compute the polynomials p ( A ) for the following polynomials (a) p ( x ) = x 2 + 2 x + 5 (b) p ( x ) = x 2 - 2 SOLUTION (a) Computing A 2 , we find
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Unformatted text preview: A 2 = -1 8-4 7 . Therefore, A 2 + 2 A + 5 I = -1 8-4 7 + 2 1 2-1 3 + 5 1 1 = 6 12-6 18 . Computing B 2 , we nd B 2 = -3-1 3 4 5 1 5 10 . Therefore, B 2 + 2 B + 5 I = -3-1 3 4 5 1 5 10 + 2 1-1 1 2 1 1 3 + 5 1 1 1 = 7-5-1 5 13 7 1 7 21 . 21 /september/ 2005; 22:09 27...
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This note was uploaded on 10/03/2010 for the course MATH 380 taught by Professor Gladue during the Fall '08 term at Roger Williams.

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