201f03fin[1]

# 201f03fin[1] - B U Department of Mathematics Math 201...

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B U Department of Mathematics Math 201 Matrix Theory Fall 2003 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.) Suppose A is a 4 × 3 matrix, and the complete solution to Ax = 1 4 1 1 is x = 0 1 1 + c 0 2 1 , c R (a) Find the second and the third columns of A . (b) Determine the ranks of the coefficient matrix and the augmented matrix. Give all the known information about the first column of A . Solution: (a) Let b T = [1 4 4 1]. From the particular solution when c = 0 it follows that column 2 + column 3 = b. The homogenous solution says that 2column 2 + column 3 = 0 . From these we obtain column 2 = - b, column 3 = b (b) We have dim Null( A ) = 1, which means rank( A ) = 2. Since the system is consistent, we have rank([ A | b ]) = rank( A ) = 2. Since the matrix must have two linearly independent

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