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10aAss2 - AS2/MATH1111/YKL/10-11 THE UNIVERSITY OF HONG...

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AS2/MATH1111/YKL/10-11 THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1111: Linear Algebra Assignment 2 Due date : Oct 12, 2010 before 6:30 p.m. Where to hand-in : Assignment Box outside the lifts on the 4th ±oor of Run Run Shaw Remember to write down your Name , Uni. no. and Tutorial Group number . ± Present your solutions/proofs logically and clearly . (The clarity of presentation counts in tests/examinations.) ± If you get di±culties, you are welcome to see the instructor, tutors or seek help from the help room. See \Information" at http://hkumath.hku.hk/course/MATH1111/index.html for availabilities. Part I: Hand-in your solutions 1. Let A be a square matrix of order n , adj A be the adjoint of A and r ( A ) be the number of leading one’s in the row echelon form of A . Prove or disprove each of the following statement. (a) adj A T = (adj A ) T , (b) r ( A ) = n if and only if r (adj A ) = n . (c) r ( A ) = 0 if and only if r (adj A ) = 0. 2. For any x ; y in R n and for any scalar ± , de²ne the operations: Addition : x ± y = x ² y Scalar multiplication : ± ² x
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