2008 Exam 2 Spring

# 2008 Exam 2 Spring - m - -- -- Page 2 of? ‘1 '1 ﬂ \ 1/...

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Unformatted text preview: m - -- -- Page 2 of? ‘1 '1 ﬂ \ 1/ Problem 1(6 pts total) " l 3 ﬁ 1 ‘1 g A. (3 pts) Find the equation of the line through the points (2,4,1) and (6,6,5) ‘l E; :4: _ / r r ‘ ‘71 a T dL'6J’FJi)"-r‘?143’) 5’ (fl-i710 ‘ Problem 3. (10 pts) Veda, ' _ If a matrix & has the following two eigenvalue/eigenvalue pairs, ﬁnd the matrix A 9:???” 1 l a0 :JL l [ 3 —1 6 +1 "‘ 3 ‘0'53 1!. H [2] ' ,, _ .- l5 :— rg Ill "3 1 O FQ l' - Z ‘—__ H i {,1 . ]+_L ’2 #5 9/2 7 2 my 5 _ l ’95“- J, , M «a J» a =1 3 9 1 lg: Lil: '3- LE 2 Hi 3 J 7 ii a 7 '4 L 2 Ill 16 i 1 1 73/2 72] A ~ ' ’ 1C1 - we; r r ‘15. d WI;— L’ﬁ-ll 39]L21[9 2 A 3) 7L5; -8 I «3 *2 ~§+e j 7 m 3 A. gé :f/ 1 .f ' 3 3 H 'j Problem 4.(10pts) Determine whether—or ~not the following three vectors are linearly independent; State clearly your conclusion and the reason for it. V7 2 W_ 0 p_ I 2 4 3 J " » ' 9- O I f 1 ’ O "/2 J '1 52—5. 2 '1 O O 3 ii. 2 o I @ﬂ 0 o o Leg (9 1 /.2 L 2 q 3 the“ o a Z =i Xu Y1 V’s .— 1 If. 4 1 o L [/2 r _ / 0;) 3 (Ci) 1 3 Mrjknomm-g 75:31-79 0': Evy antiwar/Ow " # ' .J I! "' m’ airwaqu r3“ ("(1 {m r (311 o (if‘ﬂﬂ'ﬁ f0 r“ x] V 8% ' r. - n “i ‘ "ﬁne :0 ire.” ’ELT-f E 5” “1 v'”9-¢'--’ \1 Fem-CQL-Lﬁﬁi D .0 k. W __ .. K41 Page 3 of? T - ."1 méffq] : 06*) (ﬂ; Page4of7 Problem 5.(14pts total) A (6 pts) Find the adjoint for matrix A O O 0 ) 44141144314102”: “(Right “("33 = ["32 (3") = jam 9<[§),);(.1)(LQ Q4304] ; ’3 ’3 5 it M «Ma—0 1011110 42 o o 1 04(9,1)=[")3(©_0)3 o X‘Hﬁﬁo ] ’5’ “’4 B. (8 pts) Find the determinant of the following matrix using any legitimate method (24: 1 o *1 2 4 2 0 —2 4 IO 51 g 1 l 012 1 1 _ .I 4:211—14 so =0 ' C3 2' 0 2 4 3 2 0. 2 41 3 2 210 —1 .O._..- o o O |_ Page 5 of 7 Problem 6.( 10 pts total, 2 each)Select the best answer for each of the following and circle your choice, A. If X is a square menu; and X2 = [0], then which of the following is NOT necessaril1_t£ue? 1 (a) X is singular .. WM” ; (b) detoe =..0.u t- *5 "ax exit: [0] - the rows of X are linearly dependent ' (e) all of the above must be true 13- HA is 8 3X5 maid—X, B is a 4353 matrix, and i_s__a_5><4__inatrix, which of the following is NOT deﬁned? ,._ (a) (AAT)2 7, r; J 7 - . if 4 (I I, J. (b) (CBA)2-- 9f I; 1i - . .. _. @ (Act a a; I- 3 W (d) (BACF l (c) more than one of the above is NOT deﬁned _ A l =7 a: p C. mm matrix A is invertible, then-55"" 3' '3' .' _ -" a zero cannot be an eigenvalue of A 1'" " must be diagonable (C) A cannot have repeated eigenvalues (d) if A is real it cannot have complex eigenvalues D. If matrix é has 4 rows and 4 columns and det( A)=2,l, then the system of equations Ag = E has: 6- " - :37 (a) no solution; ‘u. a is i b) exactly 1 solution; Qt“: "t c exactly 2 solutions; y__ '- (d) an inﬁnite number of possible solutions; x’ (6) all of the above are possible. E. If matrix é has 4 rows and 4 columns and timid A )= 4, then the system of equations \$3.; : b has: (in no solution; X ' (gs) xactly 1 solution; ((1) exactly 2 solutions; (e) an inﬁnite number of possible solutions; (f) all are pessible. Problem 7.( 10 pts)if A =8 3, ﬁnd the approximation to an eigenvector after two iterations of the power method, beginning with the vector i -' ' ' 3 f ’“ V r 11 1 1 a ' ll ] I'. I T J/ 0 2‘ 5 l 1—1,; a U2 [ “01V” . ,4 ¥ II”; r xi- 3 _ 9 V” Set I V r 2 i _} :9. f “3 v: i V; If Q. "J. ‘l -' .i 'I m I k _ » 33/... WI 0 I i I” 3 i l '\ J - F —I' I ll %/3 I fl 0 c" L“-. -" l ‘l ;.. L- P d _J' r -l 'J I” ‘L J \_ J " a) Q — _\ A " I' i i AIM : iZJE i i 'r ’9' -1 ’Q/u ,, , i | 'I = f ' . I- _ ' 2 c r! 4- t“ ' .. ‘ 1‘ “H ' ' _ -- ~ In; 2 a, a Page 6 of? 1 3 3 2 r >4 Problem 8.( 15 pts total) Consider the system AX=B where A : [ 2 6 9 5] c5 3 l a K _1 —3 3 0 3.: 0 (10 pts)A..If B i [0] , What condition(s) must the components of B satisfy in order for AX = B to be 0 ' 9 , ._ conglstent. ’1 g Q 2 m 1 f I g 3 2 m I (1% 3 2 x] 23-56 0 O 3 i Him; ,3 3’.) o o 81 542*?» Joégjbl—T—4 i 0031115 jﬂwl o o e 2 33m 9. 'I ’3 3 0 3),: .— 2 \ f8." f1. 0 0 8 l 5‘21 _) O 0 T A: new“ O O O C) g‘l’ﬂ._gd2ﬁ_ H“ O _i—E.-:JT‘_\L.}’ _ wk? T " 42-! _ "é Efﬁe) (40 ecmgseeﬂ-t) \$43 #3! ram .eo E: 4283 ﬂeet/'0 git}. ﬁgam’tf/O %“Zj "Raﬁ-*0 ,1 1 (5 pts)B. .If B = [5] , ﬁnd the general form of solutions to AX=B and express your answer as a linear 5 combination of column vectors. “1 3 3 2 I Fj’gfj 2 6 9 S 5 > “I *3 :3 o 5 fﬂﬁi , 9 2 1 . t a Eff?“ o o 3 5 3 :(gLfL LC) O o 0 o 4 unclear»:- j c.) 33 ‘ \/ ' C *2 1| '3 l Ya t’ O ' Y4 o ” '3 M/ 9 '- Page 7 of? ..._....______ 2<1 4 — 6 Problem 9.(15 pts) The matrix A : f} 2 84 :5] has eigenvalues 3,4,4. Find a maximal linearly 1 a. —12 E , W ‘ Independent set of eigenvectors indicate whether A is EgegonaEa A rd a. >616“ a lea-x e; o 35% HJAL) : O Y ‘ 1 \ fl ——.__ _‘> ‘25"6 O a?" L — _ _ ZZBQ +5 J’ezwtr 6 2:5: 4? 1&4 6 0 _ a v _ '2 9—3 “(5, :‘82+( 23 (D f -2 O z 1 0 H2 r4 0 A 2 1—3 1.1-5 0 1 '2— 0 2 3 ’1 ’4 83*61— 0 6 O afbﬁﬂx Xi : 4912 )Z‘N __ Q #4) 'u ’52. O ‘ * r w G 0 “4(2ij—é7ﬂg Xa/‘u 2 y) FM A24 ,_12 CI —6 F—‘I 2 -2 to 92 3 O _2 4 '6 ff“? “'2‘ ‘5‘ “6 0 E4 HA2, 4 —e o , 2 E mzug -24_M_JHKOGOO “L E n I #2" 3 O n‘ *1 3 o 2 1 -'l ’2. “3 6 31% 10 0 0 OJ {8+ K's? 3x3 OVEN}- 0 C) 0 Cl 0 C) o 0 [ﬂ 3232—, "I‘ ‘L "-3 | X) /: ll 3/2 + G bra / O 1 We 10¢sz 099mb p4 \01‘51 we: +01” 3 5 8 a) Q \J I 2 )I7/ /' - 3 ,xaﬁ ‘ " ‘ j o 7 2 ‘1 - a J i j‘ ...
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## This note was uploaded on 04/15/2011 for the course ECE 2331 taught by Professor Barr during the Spring '08 term at University of Houston.

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2008 Exam 2 Spring - m - -- -- Page 2 of? ‘1 '1 ﬂ \ 1/...

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