Math 205B&C 03/13/09

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12 -12

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1. Let B= 6 -5 .

1A. Find the characteristic polynomialofB. Showall your steps.

C''''.r f'!J (B)" t&tW'~). -~~J) ,.('4- ).)(-S--),)+f2.

L ;' - '° +S ')..--/2.>..,.')\2.of-'f'L

.. )..'Z..- 1A.,. /2-

r...u.' £t,.. /J I0-= ~...'t)(~-J}

lB. What, if any, are the eigenvaluesof B? x'1I'J e; ~ ') w: /lit: - 'I. f'

H-'t ,,. >. -Ja..c..~.~-.t:

1C. What number m should the "6" in Bbe replaced with, so that the res mg ma nx as A= 0 as

an eigenvalue? (The other eigenvalue will be new, too). How did you find m?

frlel/,J :L: C~ m ~() &4 A to 4\ /ldJr ~2.." A= 0 ~ ti'? et~ ~jj]

1It: chAr. f1: ~ [J2i ~rr:t. bits!::

,I~;~/':;~l= -00-t'A +'A'l.+/21Y) ~ih. eoJ£ ~1-./.

.J ,d, I "- '-D"'? \, \1.+ ~~.; A. IYItl

l~jf( CJIk-

c~ose (y}:::.5 TillS ~ f9 rA+ f\ ",

J="' 'A\.'" 7~ L~4 bfc- I\IUrAj Q.Colcl~f1~J

'4 A (~-1) ~[n. -I"Z-]-= [17. -12.]

y'YI-s 5-5

,Quiz 06 page 1 ~ame~8am

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2 2 -2

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2. Let A= 1 1 2 .

1 -2 5 @r'Yl;::z.s:-

2A. It's a fact that v ~[ ~5] is an eigenvector of A. Find the eigenvalne by direct compntation of Av.

SO A~:It ).Q ,(;r J;OhK>.. :

[Z. ~-'2.

J[

-rJ[-10][S"J2>. tC:"\

z.

fh7LJ)Av::: : ..~ } ~ :: ~: ~2. f:t V; c.L; \CY

2B. It's a fact that A= 3 is an eigenvalue of A. Find a basis for its eigenspace.

Ja4( a, ft 1'niI ~~,j j, IW/ (A-1:r:)

[J

.

['-l.. l.]

/tOUl fl-.u::='- -,' -"1 ~t ;ik MfF b; ~ ~ $'/ &0

I-1.. 1-

~b.~< ~([UJn}~.rIMM ~r" kdwS~~'

2C.WhatisthedimenIDonof;heei{~;i(;&.I~@k.,j (ur'" t-, ;,t, ntJ'f-j