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: Page 1 of 8 (f
Obi 03‘ mil
Signature: Proctor: DO NOT OPEN THIS BOOKLET ‘UNTIL YOU ARE INSTRUCTED TO DO SO ECE 2331 Exam 2 Spring 2002 INSTRUCTIONS: (1)Fill in the information required on the upper left of this page. (2)1ncluding this cover page, this exam has 8 pages. Verify this now and raise your hand if you have
a problem. Do NOT look at the questions! (3)This exam is closed book and closed notes except for one page(two sides) of notes which must be
handed in with your exam. Be sure your name is on the page so it may be returned to you. ' (4)Communication devices of any kind (eg. cell phone, pager, etc.) may not be used during the exam.
Turn them off or give them to the room monitor. (5)The use of a calculator is required, but all steps must be shown. Unless otherwise indicated, round
your results to 4 signiﬁcant digits. (6)You will have 90 minutes for this exam. (7)When you ﬁnish the exam, unless the twominute warning has been given, put your crib sheet
inside the exam, hand it in and leave the room. (8)0nce the twominute warning is given, stay seated until time is called, at which point you must
stop writing immediately and stand up. Continuing to work after time is called will result in substantial
penalty. Prob. 1. / 18 Prob. 3. /20 Prob. 5. / 12 Prob. 2. /20 Prob. 4. /20 Prob. 6. /10 TOTAL SCORE %1H \1’% ‘33 57, L1)
Page20f8 Problem l.(18 pts total)Select the best answer to each of the following and circle your choice. A. Assuming that all of the indicated operations are deﬁned, which of the following is alwaystrue in matrix
arithmetic?
1. (A+B)2 = A2+ 2AB + B2
2. (AB)2 = 132A2
3. (513)T = 1/5 BT
( AB)T AT = BT(A2)T
5. A(AB2—AC2)=(ABAC)(AB+AC)
6. all of the above are true B. Which of the following systems AX = B could be inconsistent?
1. 3 equations in 4 unknowns with rank(A) = 3
® 4 equations in 3 unknowns with rank(A) = 3
5 equations in 5 unknowns with rank(A)=rank([A B]) = 3
6 equations in 4 unknowns with B= [0]
each of the above could be inconsistent
none of the above could be inconsistent P‘P‘PS” x=2—t
C. The parametric equations y = 4 + 3t describe a z=1+2t 1. line perpendicular to the vector (1,3,2) 2. line parallel to the vector (2,4,1) 3. plane perpendicular to the vector (1,3,2) line parallel to the vector (1,3,2)
5. plane through the point (2,4,1) D. A is a 5x7 matrix of rank 5, then
the r0ws of A are linearly independent
2. the columns of A are linearly independent
the dimension of the solution space Ax=[0] will be 5
the equation AX=I cannot have solution
all of the above are true SAFE” E. If A is an nxn matrix, which of the following is NOT true about det(A)?
1. det(AT) = det(A)
@ det(ocA) = adet(A) for all scalars 0L
3. det(A2) = det(A)det(A)
4. det(A'1)det(A) = 1 if det(A);é0 F. If A is an nxn matrix with the second row of A equal to twice the ﬁrst row of A, then
1. det(A) = 2
2. rank(A) = 2
CS A is not invertible
4. rank(A) = 0 Page 3 of 8 G. 1.434;)... mac.) 1. 6 is an eigenvalue of A
2. 4 is an eigenvalue of A
@ 2 is an eigenvalue of A
4. 3 is an eigenvalue of A
5. none of the above is necessarily true H. The strongest statement we can make based on Gerschgorin concerning a lower bound for the real 3 4 2 __ 4 lX~3\g¢ 333
eigenvaluesofthematrix[l 2 —l]isthat ‘A‘Bxéc’ 3“" Jigs; HZlég J3<_A dag 5—11 lX*L\€?' 0%) A ‘3 “c3 27> “26—2
l. allrealeigenvaluesareZO RAF=4» 45$) \ ‘ ~ . J 2. all real eigenvalues are 2 2 4 )
(Q all real eigenvalues are 2 —3 ‘3 '
4. all real eigenvalues are 2 S
5. all real eigenvalues are 2 6 1 200
1. Suppose P‘AP= 0 3 0 .Then eA is
001
.4
200 w
1. P0 3 0P"l BANK
001
200
P[030)P“
2_ e 001
e2 0 0
3 P" o e3 0P
0 o 1 Page 4 of 8 Problem 2 (20 pts)Use Gaussian elimination on the augmented matrix to solve the system below. If
there is no solution, indicate clearly why. If there are an inﬁnite number of solutions, express the
general form for solutions as a linear combination of column vectors. Use EXACT arithmetic and show
all steps! 2x1+x2—x3+4x4=6
x1+x2+x3+x4,=4
3x1+2x2+5x4=10 2x +2x +2x +2 =8 1 '\ ‘1
1 2 3 x4 gait ‘  ‘ ‘ q “till. 1 \ 2,
\\\\”§ ‘3‘ gg~32~l —> 0\3~'L
'2 “" q k Rytuk 2' “.‘ W E . 0" '3 Z _7—— 0.4.32UZ.
\\l\‘( 103/“) 233“) 00300
3209‘; $112.2.9 Rug 000°C
21?. Z “V o \‘12‘1.L 3 *2!
o c Q 0 0 XL: 1. x3 ‘1 X “ ﬁstIX; F3“
0  (14" *‘X‘BJ’V H
o o o o x\_  3
—Z
2 Z.
Xi Z «'3
X
XL. .. 1' *X" l ‘k ‘ 0
O
Xw Page 5 of 8 (6 —2 1\  Problem 3(20 pts)The eigenvalues of the matrix A = —2 9 —2 are A = 5,5,11. Find bases for each
1 —2 6
of the distinct eigenspaces and indicate whether or not A is diagonable. QAA‘S‘ i ~‘¢ I 0 RLHQS l L \O {thxme — ~ ——~> i ~>.i o (LAM 0 o’ o a' (2; g: ;L(:BH,G‘
m at M133 % ’A; W‘ 1 :3, :1; :LOO ‘ 33:: (32,:2119100 ELR, \ 0:: :51100>~WQL 0: :00
\ 1—5 0 5 7~ ‘ 0 RM“ ‘5 ‘n 1“ ° 8 “‘24”
Wm.» g L: we;
—~> 0 \ z OXX11'1X3
° ° ° 0 mm 9M Page 6 of 8 Problem 4. (20 pts) Suppose A: LU , where 100 14—2
L=310U=013
121 001 \.'.\ ‘1‘
L\M(= t)” => DK 1 “x 21> 3:;wa ”’3 ”A7,:\3'\= ‘7—
\ l __ 7&37'
_. a ' ‘f.\\—\\yL~LX3 ~— ~ .
uX~\\' ((3 ‘3 XL¥3K31'1 :9 71223L13~ P1 6g
39 1‘1 x, = \—\( Lxﬁ MA =
x. £5
\(1 q ~H
v‘ Page 7 of 8 1131 1131 1131
1—123 122—1 2142
Problem5.(12ptstotal)LetA=1 _15 7 3:1 —1 5 7, C=1 _1 5 7
abcd abcd abcd
1131 2262 113 1
_1—157 _1—123 _01—1—2
D‘1.—123’E“1—157’F‘1—157
abcd abcd ab c (1 Suppose detA = —4 and det B = —20 . Find each of the following (Hint: see how to get the new matrix from A
and/or B)  ~ 12
1. detC 0W up); “WW vim, zAkaQ .valew
23 MC: MRkMB= ~‘Hlo: §1_ Lee“ 2 2 = “1
2. detD OW “K” 6 :'—> M9" M“ ‘— _ —‘8
. up“, EzLMR=9~L%'/~
3. detE QM Rash.9 M 4. detF BM 6"» Page 8 of 8 Problem 6. (10 pts) Suppose A is a 2 x 2 matrix with Tr(A) = 3 and det A = 4 . Find the two eigenvalues of A. AmazxWiﬁﬁ 1») 2pm 3‘31: (N‘QRBV‘ :.:.=C3;A":))‘L1 Ag :5 3'),j>i_%“‘\t 0
=3 )1 ~31L~“\ .. 0 ...
View
Full Document
 Spring '08
 BARR

Click to edit the document details