Eigenvalues - Linear Algebra - Quiz Solution, Exercises of Linear Algebra

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Math 205B Quiz 07 page 1 03/12/2010 Name. Susgasied salhion 1. Suppose that =| 9, oF 1A. Find all values of k for which M has \ = 3 as an eigenvalue with multiplicity 2. Show all your work. =e a a = (o-r)(6-»)tK = \- 6d 4K a 2 HERE we reguive. \*-6rX +k = (-3) » 1B. Find all values of k for which M has eigenvalues 5 and 1. Show all your work. Mo ue feguine Hat '-Gd+K = O-H>-9, tach 6 N-EN*S 1C. Find all values of k for which M has no real eigenvalues. Show all your work. th rob f N=G>*K in general aoe + /%-4K z= Get ay qt at. ea > 36-K< O S %< MK 2 8 -3 2 2.LetA=| 2 1 4 |. Here are three facts about A: Fact 1) One eigenvalue of A is a = 7. -5 15 -3 1 Fact 2) The vector v= | 2 | is an eigenvector of A for some other eigenvalue 6 # 7. —5 Fact 3) In no part of this question do you need to use determinants. 2A. Use fack 2 to find that other eigenvalue 8. Show all your work. ( -§ ( A |= “he --«(s]>@-9) s ce) 2B. Find a basis for the eigenspace for the eigenvalue 7. Show all your work. pref (A-72) = ref E 7 ‘); fe | > a 7 tes HE (3 basis 0 bedter) 2C. Find the characteristic polynomial of A. pote vel, SCE. te ign 4 a ie 2, fhe mabey of ? = ee fan (>=?) ml (48) rat ton 6 hor 4 cher spa 4p ol Ae IBD digree of “4 dhe pay 6 3. THE Uy porsboily 5 i (- Aang (2% pethops be Ke Li = )