fm241sol2 - A = 8; AB T = 6 6 2 2 ; BA T = 6 2 6 2 ....

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M241: Matrix Algebra Homework Assignment 2 Answers Graded Problems: Section 1.5 Problems 15, 28, 32; Section 1.6 Problems 6, 36, 50. Section 1.5. Problem 15. The PA = LDU factorizations: 0 1 0 1 0 0 0 0 1 0 1 1 1 0 1 2 3 4 = 1 0 0 0 1 0 2 3 1 1 0 0 0 1 0 0 0 - 1 1 0 1 0 1 1 0 0 1 ; 1 0 0 0 0 1 0 1 0 1 2 1 2 4 2 1 1 1 = 1 0 0 1 1 0 2 0 1 1 0 0 0 - 1 0 0 0 0 1 2 1 0 1 0 0 0 0 . Problem 28. The A = LDU factorizations: A = ± 2 4 4 11 ² = ± 1 0 2 1 ² ± 2 0 0 3 ² ± 1 2 0 1 ² ; A = 1 4 0 4 12 4 0 4 0 = 1 0 0 4 1 0 0 - 1 1 1 0 0 0 - 4 0 0 0 4 1 4 0 0 1 - 1 0 0 1 . Problem 32. Lc = b gives c = ± 2 3 ² , and Ux = c gives x = ± - 5 3 ² . Check that Ax = b . Section 1.6. Problem 6. A - 1 1 = 1 0 0 - 1 1 - 1 0 0 1 ; A - 1 2 = 3 / 4 1 / 2 1 / 4 1 / 2 1 1 / 2 1 / 4 1 / 2 3 / 4 ;
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A - 1 3 = 0 - 1 1 - 1 1 0 1 0 0 . Problem 13. A T B = 8; B T
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Unformatted text preview: A = 8; AB T = 6 6 2 2 ; BA T = 6 2 6 2 . Problem 36. A-1 = 3 / 4-1 / 2 1 / 4-1 / 2 1-1 / 2 1 / 4-1 / 2 3 / 4 . Problem 40. (a) True. Gauss elimination will produce at most three pivots; (b) False. Counterexample: A = 1 1 1 1 is not invertible; (c) True. Since AA-1 = A-1 A = I , A is the inverse of A-1 ; (d) True. Since ( A T )-1 = ( A-1 ) T , A is invertible. Problem 50. AB = BA if and only if ( AB ) T = ( BA ) T , which is equivalent to B T A T = A T B T . Q.e.d. 2...
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fm241sol2 - A = 8; AB T = 6 6 2 2 ; BA T = 6 2 6 2 ....

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