2331_Test_3A_sol_Sp11 - Math 2331 Test 3 A Page 1 of 8 Math...

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Unformatted text preview: Math 2331 Test 3 - A Page 1 of 8 Math 2331 Linear Algebra Spring 2011 Test 3 Version A Last Name , «- . M First Name myUH# No Notes No Books No Calculator The point values for each problem are indicated on the test. Partial credit may be awarded for substantially correct work, but only if you show your work AND the grader can follow the order of the solution steps. (3 points each) Question 1: Given the matrix A below, and assuming det(A) = —5, C f I as H Crqu D‘FDD" /~'\ \\ (a). Is A invertible? mm) 1% 0 Find the determinants of the following matrices: / (b). det((A‘1)T)= .; J... Math 2331 Test 3 ~ A Page 2 of8 a Azd g «)wpmk (;Q]3&Lg/};.wg(;st#EEE9 ' a b c / _ A "\i (d). det[—4d —46 41f] :1 “’ % (£52k W @ g 12 1' Z]: 1:: (10 points) Question 2: Use Cramer's Rule to solve the following system for y: _ZX+ 4y+ 2:3 ,— Q Lf- I ‘ X 7 X+ y+ Z: l l t 3 3X— y~ 32:10 3 “I __ 3 g {:2 3 I , , 81:2 IO 1 &&é30+7+M7 3 [0 <3 «0430)»(1—9) 2 #8 MP! 2 é+/;>+ -~/-— 3 ~Q-~€~zgfl : oak &&B;_‘%g.‘(fi V v 5! : MR 52% 3030‘ C . f 1' ERIC“ C (e). det f 1' Math 2331 Test 3 - A Page 3 of8 (15 points) Question 3: Find the determinant of the following matrix: 12030 3 5040 M2—2-4330 0 00~20 3 6325, r» V 5+5 1610 3 Mm: SCSS’: M 350% r— 5072) (3) C33 v 1 f : SGQX’BD Mtéé : 56—93 (37 (5’9 ’2 BO Math 2331 Test 3 — A Page 4 of 8 (10 points) Question 4: Given the following vectors: with . A: 8 [< (a).F1nd ux W 1-, [ Kg) ( 9. 3 O = (MN “(04233 +(I’>-oll< ‘3— "3i +97% 43k (b). Find wx W (15 points) Question 5: Given the matrix M below 1 Ho ] 3 (a).Find the determinantofM "I" O ~— 3 6 .__O 2 «it; (b). Is M invertible? W (C). State the sum of the eigenvalues of M. ‘1’ Liz + 5 I; / O 1 {713.921 0,50 0100-) ((1). State the product of the eigenvalues of M. \ 2W”; “[6 10 3 m:(e%o> 305.[ _ M £5 0 A CH “’("q 0 5i " ‘QO +2 ; t ,. O _. A 4 {+3 0 _ g, 1 C13'(0 /:’> <27" / s 1“ O 3 : Can/(4 /05,/ G v t: _ 24'} [3 :: SfiCf: “Q; Ca; (flfijgsj C523: (“0 fé‘éll 0 3+! C33 _. (13$ ‘2 “D 1460/ ” "/1 \31—2 { x x ' 3 l ’ [O M (53 (“(73+3/o%} " [71 C 2 $20 0‘ 4:2 CTIC O *4 O ~a 0 LP , L 1 J— '-”?{L( MM““!Q m” ;~(£°C” ”° Math 2331 Test 3 — A (e). If M is invertible, use cofactors to find the inverse of M. Page 5 of 8 Math 2331 Test 3 — A Page 6 of 8 (10 points) Question 6: Are the following vectors eigenvectors for the matrix 1—2 1 3 4 ~l A: O 0 1 2-10 2 ~l l (a).v——1 ~l l l b“: ()’(1 l (10 points) Question 7: State the Characteristic polynomial for the following matrix. Note that the factored form is sufficient. 1 4 5 Lil S“, j WW: :(o -3~A ~g O O Q~>x P m: (Sm {A a) I) r” (We) 63 a} 569/1) Math 2331 Test 3 — A Page 7 of 8 (15 points) Question 8: Find the eigenvalues and corresponding eigenvectors for the following matrix: A=25 9—09 5' [41) “"1I;(r we) cm (Pr/11) ; (9000417 ~ £10 0 : 76-31 ~48 (wwiosm) M ,— 1,262 Math 2331 e Test 3 — A Page 8 of 8 (10 points) Bonus: Given any 3x3 matrices A and B, are the following statements true or false? If a statement is false, explain why or give a counterexample. (a). det(BABT) = det(/—l) {:31} L5 5 MGQABT): all: e a clef A a eels 6 #cst/él (b). If A is singular, then AB and BA are also singular. TKUE M ft 2*» O MCMFM B‘Mfl;6 % S’W/ MmellwfloM/BSO ’ ’ (e). The eigenvalues of A + B are simply the sum of the eigenvalues ofA plus those of B. We Fargo/s2. (a 3 321. (d). If M is a 2x2 matrix with eigenvalues 3 and —5, find the eigenvalues of M + I . my? 6% W -?VCW a”) l 5 (“4’1 3 a: b é\‘ / , OL’CR'J) A ~_~, a?“ A Celt/<7 MW +1611) :0 Met 20»: “By-'5’ (mu/karat?) M: “l4 Ail/t“? 3+/,:<t ...
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