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201s06fin

# 201s06fin - B U Department of Mathematics Math 201 Matrix...

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B U Department of Mathematics Math 201 Matrix Theory Spring 2006 Final Exam This archive is a property of Bo˘gazi¸ci University Mathematics Department. The purpose of this archive is to organise and centralise the distribution of the exam questions and their solutions. This archive is a non-profit service and it must remain so. Do not let anyone sell and do not buy this archive, or any portion of it. Reproduction or distribution of this archive, or any portion of it, without non-profit purpose may result in severe civil and criminal penalties. 1. Prove: If the entries in each row of an nxn matrix A add up to zero, then det ( A ) = 0. (Hint: Consider the product AX where X is an nx1 matrix, each of whose entries is one) (20 points) Solution: For A = a 11 a 12 · · · a 1 n . . . a n 1 a n 2 · · · a nn and X = 1 1 . . . 1 AX = a 11 + a 12 + · · · + a 1 n a 21 + a 22 + · · · + a 2 n . . . a n 1 + a n 2 + · · · + a nn = 0 0 . . . 0 (Since the entries in each row of A add up to zero) So X = 1 1 . . . 1 = 0 is a solution of AX = 0 det ( A ) = 0. (Recall: If det ( A ) = 0, then AX = 0 has only the trivial solution, X = 0). 2. Find the equation of the best line through the points (-1,-2), (0,0), (1,1) and (2,3). (25 points) Solution: Let y = C + Dt be the best line fitting the given data. Then for A = 1 1 1 0 1 1 1 2 , b = 2 0 1 3 and X = C D ( A T A ) x = A T b must hold. A T A = 1 1 1 1 1 0 1 2 1 1 1 0 1 1 1 2 = 4 2 2 6 A T b = 1 1 1 1 1 0 1 2 2 0 1 3 = 2 9 4 2 2 6 C D = 2 9 i

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4 C + 2 D = 2 2 C + 6 D = 9 D = 1 . 6 , C = 0 . 3
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201s06fin - B U Department of Mathematics Math 201 Matrix...

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