Winter 2007 - Tsogtgerel's Class - Exam 2

Winter 2007 - Tsogtgerel's Class - Exam 2 - MATH 20F WINTER...

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Unformatted text preview: MATH 20F WINTER 2007 MIDTERM EXAM II FEBRUARY 28 Problem 1: Let B be a 3 3 matrix such that det B = 2, and let A be given by A = 2 4 1 3 9 1 8 64 1 13 169 3 5 : a). Calculate det( BAB ). b). Write A in LU form, that is, nd an upper triangular matrix U and a unit lower triangular matrix L such that A = LU . Solution: 1a ). A property of determinant implies det( BAB ) = (det B )(det A )(det B ) = (det B ) 2 det A = 4 det A; so if we know the determinant of A we can calculate the determinant of BAB . Let us use row reduction to nd det A . 2 4 1 3 9 1 8 64 1 13 169 3 5 2 4 1 3 9 5 55 10 160 3 5 2 4 1 3 9 5 55 50 3 5 : Taking the product of the diagonal entries of the echelon form, det A = 1 5 50 = 250, and so det( BAB ) = 4 250 = 1000. 1b ). In the above row reduction, we used only row replacements, and the multiplying numbers we used were 1 , 1 , and 2 . We can take U equal to the above echelon form and construct L from the (negatives of the) multiplying numbers as follows. U = 2 4 1 3 9 5 55 50 3 5 ; L = 2 4 1 1 1 1 2 1 3 5 : 1 2 MATH 20F WINTER 2007 MIDTERM II Problem 2: Let the matrix A and the vectors u 2 R 4 and w 2 R 3 be given by A = 2 4 3 2 4 4 2 6 2 4 2 3 9 3 5 ; u = 2 6 6 4 14 10 1 14 3 7 7 5 ; w = 2 4 23 43 3 5 : a). Find a basis for and the dimension of Nul A . Is u in Nul A ? ( Hint on row reducing A : Scale the second row by factor 1 2 and interchange the rst two rows. Then do not use scaling until the last moment. This will save some arithmetics on fractions.) b). Find ab)....
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This note was uploaded on 04/23/2008 for the course MATH 20F taught by Professor Buss during the Winter '03 term at UCSD.

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Winter 2007 - Tsogtgerel's Class - Exam 2 - MATH 20F WINTER...

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