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08aAss1

# 08aAss1 - 4 A square matrix A is called a nilpotent if A r...

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AS1/MATH1111/YKL/08-09 THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1111: Linear Algebra Assignment 1 Due date : Sept 22, 2008 before 6:30 p.m. Where to hand-in : Assignment Box outside the lifts on the 4th ﬂoor of Run Run Shaw Remember to write down your Name , Uni. no. and Tutorial Group number . If you ﬁnd diﬃculties, you are welcome to see the instructor, tutors or seek help from the help room. See “Information” at http://147.8.101.93/MATH1111/ for availabilities. Normally we do not count assignment grades in your ﬁnal score. Nevertheless, any discovered plagiarism will be referred to University Disciplinary Committee. 1. Find a , b and c so that the system x + ay + cz = 0 bx + cy - 3 z = 1 ax + 2 y + bz = 5 has the solution x = 3, y = - 1, z = 2. 2. Let A and B be symmetric matrices. Prove that AB = BA if and only if AB is symmetric. 3. Let A be a square matrix of order n . Prove the following: (a) If A is invertible and AB = 0 for some n × n matrix B , then B = 0. (b) If A is not invertible, then we can ﬁnd an n × n nonzero matrix B such that AB = 0. [Here 0 means a zero matrix of an appropriate size.]

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Unformatted text preview: 4. A square matrix A is called a nilpotent if A r = 0 for some natural number r . (a) Let A = ( a ij ) n × n be a matrix whose entries satisfy a ij = 0 if i ≥ j . Show that A is a nilpotent. (b) Is there a nilpotent matrix such that all of its entries are nonzero? Justify your answer. 1 5. Let A = ± 1 2-1 1 ² and C = ±-1 1 2 1 ² . (a) Find elementary matrices E 1 and E 2 such that C = E 2 E 1 A . (b) Show that there is no elementary matrix E such that C = EA . 6. In each case ﬁnd an invertible matrix U such that UA = R in reduced row-echleon form and express U as a product of elementary matrices. (a) A = ± 1 2 1 5 12-1 ² . (b) A =   2 1-1 0 3-1 2 1 1-2 3 1   . 7. Prove that B is row equivalent to A if and only if there exists a nonsingular matrix M such that B = MA . 8. ( Harder ) Let A be an m × n matrix, and B be n × m . Prove that if n < m , then AB is not invertible. 2...
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08aAss1 - 4 A square matrix A is called a nilpotent if A r...

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