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201f04mt1[1] - B U Department of Mathematics Math 201...

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B U Department of Mathematics Math 201 Matrix Theory Fall 2004 First 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. Prove the following statements: (a) Let A and B be symmetric matrices. If AB is also symmetric then AB = BA . Solution: A and B are symmetric means A T = A and B T = B . Also ( AB ) T = AB . Now using all these: AB = ( AB ) T = B T A T = BA . Proof is done. (b) If AB = BA and B is invertible then AB - 1 = B - 1 A . Solution: We are given AB = BA and B is invertible. Multiply this identity from both sides by B - 1 to obtain: B - 1 ABB - 1 = B - 1 BAB - 1 ⇐⇒ B - 1 A = AB - 1 . Proof is done. 2. Let A = 1 0 1 0 1 - 1 1 2 0 and B = 1 0 0 1 1 - 1 0 1 0 . (a) Show that A and B are invertible matrices by finding their inverses explicitly. Solution: We construct the augmented matrices: [ A : I ] and [ B : I ] and apply elementary row operations: [ A : I ] = 1 0 1 1 0 0 0 1 - 1 0 1 0 1 2 0 0 0 1 e 1 : - r 1 + r 3 r 3 ---------→ 1 0 1 1 0 0 0 1 - 1 0 1 0 0 2 - 1 - 1 0 1 e 2 : - 2 r 2 + r 3 r 3 ---------→ = 1 0 1 1 0 0 0 1 - 1 0 1 0 0 0 1 - 1 - 2 1 e 3 : - r 3 + r 1 r 1 ---------→ e 4 : r 3 + r 2 r 2 1 0 0 2 2 - 1 0 1 0 - 1 - 1 1 0 0 1 - 1 - 2 1 . Hence we have found that: A - 1 = 2 2 - 1 - 1 - 1 1 - 1 - 2 1 .
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[ B : I ] = 1 0 0 1 0 0 1 1 - 1 0 1 0 0 1 0 0 0 1 f 1 : - r 1 + r 2 r 2 ---------→ 1 0 0 1 0 0 0 1 - 1 - 1 1 0 0 1 0 0 0 1 f 2 : - r 2 + r 3 r 3 ---------→ = 1 0 0 1 0 0 0 1 - 1 - 1 1 0 0 0 1 1 - 1 1 f 3 : r 3 + r 2 r 2 --------→ 1 0 0 1 0 0 0 1 0 0 0 1 0 0 1 1 - 1 1 . Hence we have found that: B - 1 = 1 0 0 0 0 1 1 - 1 1 . Since A and B are row equivalent to the 3 × 3 identity matrix, they are invertible matrices. (b) Express A and B as a product of elementary matrices (Do not perform explicit matrix multi- plication, but perform inversions, transpositions etc. whenever necessary).
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