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final

# final - Linear Algebra S2010D Sec.1 Summer 2006 Final Exam...

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Linear Algebra S2010D Sec.1, Summer 2006 Final Exam Name: June 27, 2006 Do all problems, in any order. Show your work. An answer alone may not receive full credit. No notes, texts, or calculators may be used on this exam. Problem Possible Points Points Earned 1 15 2 10 3 15 4 15 5 10 6 10 7 15 8 10 TOTAL 100 1

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1. (15 points) Given A = 3 2 2 3 (a) Explain what it means for a symmetric matrix B to be positive definite . (b) Is A positive definite? Explain. (c) Find a matrix B such that B 2 = A or explain why no such matrix exists (you need not simplify your answer). (d) Find a matrix C such that e C = A or explain why no such matrix exists (you need not simplify your answer). 2. (10 points) Recall that a square matrix A is called nilpotent if there exists a positive integer n such that A n = 0. (a) Show that the eigenvalues of a nilpotent matrix are zero. (c) Give an example of a nonzero nilpotent matrix. 3. (15 points) Given the singular value decomposition A = U Σ V T with U = 1 3 - 1 2 2 2 - 1 2 2 2 - 1 Σ = 4 0 0 0 0 1 0 0 0 0 0 0 V = 1 2 1
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