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Unformatted text preview: Linear Algebra S2010D Sec.1, Summer 2006 Exam 1 Name: June 9, 2006 Do all problems, in any order. Show your work. An answer alone may not receive full credit. No notes, texts, or calculators may be used on this exam. Problem Possible Points Points Earned 1 12 2 10 3 12 4 12 5 5 6 5 7 12 8 10 9 10 10 12 Extra Credit 10 TOTAL 100+10 1 1. (12 points) Find all solutions to the follwing system of linear equations x y 2 z 8 w = 3 3 x 2 y 3 z 15 w = 9 2 x + y + 2 z + 9 w = 5 Solution: Using row reduction, we find that the RREF of the matrix 1 1 2 8 3 3 2 3 15 9 2 1 2 9 5 is 1 0 0 1 2 0 1 0 3 3 0 0 1 2 1 so the solution set is the set of all  2 3 1 + t 1 3 2 1 for t in R . 2. (10 points) (a) Define lower triangular matrix. (b) Show that if A and B are 3 × 3 lower triangular matrices then AB is also lower triangular. (c) Suppose A is a 3 × 3 lower triangular matrix with nonzero entries along the main diagonal. Show that A is invertible with lower triangular inverse. Solution: (a) A square matrix A = ( a ij ) is called lower triangular if a ij = 0 whenever j > i . (b) Write A = a 11 a 21 a 22 a 31 a 32 a 33 and B = b 11 b 21 b 22 b 31 b 32 b 33 . Then AB = a 11 b 11 a 21 b 11 + a 22 b 21 a 22 b 22 a 31 b 11 + a 32 b 21 + a 33 b 31 a 32 b 22 + a 33 b 32 a 33 b 33 which is lower triangular....
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This note was uploaded on 10/18/2011 for the course S 2010D taught by Professor Peters during the Spring '10 term at Columbia.
 Spring '10
 Peters

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