This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 2331 Test 3  A Page 1 of 8 Math 2331
Linear Algebra
Spring 2011
Test 3
Version A
Last Name , Â«
. 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) = â€”5, C
f I as
H
Crqu
Dâ€˜FDD" /~'\ \\ (a). Is A invertible? mm) 1% 0 Find the determinants of the following matrices:
/ (b). det((Aâ€˜1)T)= .; J... Math 2331 Test 3 ~ A Page 2 of8 a
Azd
g Â«)wpmk (;Q]3&Lg/};.wg(;st#EEE9 ' a b c / _ A "\i
(d). det[â€”4d â€”46 41f] :1 â€œâ€™ % (Â£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 ,â€” Q Lf I â€˜ X 7
X+ y+ Z: l l t 3 3Xâ€” y~ 32:10 3 â€œI __ 3 g {:2 3 I , ,
81:2 IO 1 &&Ã©30+7+M7
3 [0 <3 Â«0430)Â»(1â€”9)
2 #8
MP! 2 Ã©+/;>+ ~/â€” 3 ~Q~â‚¬~zgï¬‚
: oak
&&B;_â€˜%g.â€˜(ï¬ V v 5! : MR 52% 3030â€˜ C
. f
1' ERICâ€œ 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
M2â€”24330
0 00~20 3 6325, rÂ» V 5+5 1610 3 Mm: SCSSâ€™: M 350% râ€” 5072) (3) C33 v 1
f : SGQXâ€™BD MtÃ©Ã©
: 56â€”93 (37 (5â€™9 â€™2 BO Math 2331 Test 3 â€” 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 â€œ(04233 +(Iâ€™>oll<
â€˜3â€” "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 ~â€” 3 6 .__O 2 Â«it; (b). Is M invertible? W
(C). State the sum of the eigenvalues of M. â€˜1â€™ Liz + 5 I; / O
1 {713.921 0,50
0100) ((1). State the product of the eigenvalues of M. \ 2Wâ€; â€œ[6 10 3
m:(e%o>
305.[
_ M Â£5 0 A
CH â€œâ€™("q 0 5i " â€˜QO
+2 ;
t ,. O _.
A 4 {+3 0 _ g, 1
C13'(0 /:â€™> <27" /
s 1â€œ O 3 :
Can/(4 /05,/ G
v t: _ 24'} [3 :: Sï¬Cf: â€œQ;
Ca; (flï¬jgsj
C523: (â€œ0 fÃ©â€˜Ã©ll 0
3+! C33 _.
(13$ â€˜2 â€œD 1460/ â€ "/1
\31â€”2 { x x
' 3 l â€™ [O M
(53 (â€œ(73+3/o%} " [71
C 2 $20 0â€˜ 4:2 CTIC
O *4 O
~a 0 LP ,
L 1 Jâ€” 'â€?{L(
MMâ€œâ€œ!Q mâ€ ;~(Â£Â°Câ€ â€Â° Math 2331 Test 3 â€” A (e). If M is invertible, use cofactors to find the inverse of M. Page 5 of 8 Math 2331 Test 3 â€” A Page 6 of 8 (10 points) Question 6: Are the following vectors eigenvectors for the matrix 1â€”2 1 3
4 ~l
A: O
0 1
210 2
~l
l
(a).vâ€”â€”1
~l
l
l
bâ€œ:
()â€™(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 Sâ€œ,
j WW: :(o 3~A ~g
O O Q~>x P m: (Sm {A a) I) râ€ (We) 63 a} 569/1) Math 2331 Test 3 â€” A Page 7 of 8 (15 points) Question 8: Find the eigenvalues and corresponding eigenvectors for the
following matrix: A=25 9â€”09 5'
[41) â€œ"1I;(r we)
cm (Pr/11) ; (9000417 ~ Â£10
0 : 7631 ~48
(wwiosm) M
,â€” 1,262 Math 2331 e Test 3 â€” 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(/â€”l) {:31} L5 5
MGQABT): all: e a clef A a eels 6 #cst/Ã©l
(b). If A is singular, then AB and BA are also singular.
TKUE M ft 2*Â» O
MCMFM Bâ€˜Mï¬‚;6 % Sâ€™W/
Mmellwï¬‚oM/BSO â€™ â€™
(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 â€”5, ï¬nd the eigenvalues of M + I . my? 6% W ?VCW aâ€)
l 5
(â€œ4â€™1 3 a: b Ã©\â€˜
/ , OLâ€™CR'J) A ~_~, a?â€œ A Celt/<7
MW +1611) :0 Met 20Â»: â€œBy'5â€™
(mu/karat?)
M: â€œl4 Ail/tâ€œ? 3+/,:<t ...
View
Full Document
 Spring '08
 Staff
 Linear Algebra, Algebra

Click to edit the document details