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201s04mt1[1]

# 201s04mt1[1] - B U Department of Mathematics Math 201...

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B U Department of Mathematics Math 201 Matrix Theory Spring 2004 Second Midterm 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. For which value(s) of the real number k, does the following linear system has: L 1 : x + y - z = 1 L 2 : 2 x + 3 y + kz = 3 L 3 : x + ky + 3 z = 2 (a) a unique solution (b) no solution Solution: First we find - 2 L 1 + L 2 - 2 L 1 : - 2 x - 2 y + 2 z = 2 L 2 : 2 x + 3 y + kz = 3 Adding side by side we get - 2 L 1 + L 2 : y + ( k + 2) z = 1 Next we find - L 1 + L 3 - L 1 : - x - y + z = 1 L 3 : x + ky + 3 z = 2 Adding side by side we get - L 1 + L 3 : ( k - 1) y + 4 z = 1 So we have L 1 : x + y - z = 1 2 L 1 + L 2 : y + ( k + 2) z = 1 - L 1 + L 3 : ( k - 1) y + 4 z = 1 Then we compute ( k - 1)(2 L

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201s04mt1[1] - B U Department of Mathematics Math 201...

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