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Unformatted text preview: 501.0170 N3 MATH 115
. Tuesday October 18, 2005 NAME: Please Print Id. 1%.: Faculty of Mathematics
University of Waterloo MIDTERM EXAM Fall 2005 7:00 — 9:00 Instructions: 1. No calculators are permitted. 2. The exam has 9 problems. 3. Make sure to put your own name and
ID number at the top of this page. 4. If you do not have enough space to ﬁn
ish your solution to a. problem, continue
it on the back of the page. and indicate
that you have done so. 5. There is 1 blank page at the end of the
exam that can be torn off and used for
rough work. This exam has 11 pages
in tote], inciuding the cover and one blank page. 6. CIRCLE YOUR SECTION AND
INSTRUCTOR BELOW: Sec. Instructor Time
001 B. Ferguson 8:30
002 R. Moose 10:30
003 B. Ferguson 8:30
004 P. Wood 2:30
005 P. Wood 3:30
006 F. Dunbar 12:30
007 F. Dunbar 8:30
008 W. Kuo 10:30
009 W. Kuo 8:30
010 R. Andre 1:30 MATH 115, Midterm Page 2 of 11 ' Name: [8] 1. Find the general solution to the following system 4x1+ $2+8$3 + 2:4 =12
321 + 922+6$3 + 22:4 = 8 w/ 41?]
31612 R; '9 [.2
a" Rtﬁzf
"! I o a
A .
3 I c .1 ‘ I O Q 
—9 l ,L/ M/
O I. o 5' ‘9
ﬁL—JK‘L—bk‘
9L 91L er. gosh.
I ‘f
:13 are. (:‘r‘t
g £ 13.25 #3: “//
=4 ISLL: ——‘{ 5’t'r
‘ 0‘1 ‘1'“: '3’ L! '
SI: (‘13) : (*VJe z/Ml >4L/
33. S o
’7 t a ) 1.,
€— ) MATH 115, Midterm Page 3 of 11 Name: [8] 2. Suppose A, B and C are all 4 x 4 matrices such that det(A) = 0,det(B) = —% and det(C) = 2.
Find ta)det(ABC) : 541A méaéE‘ouc
= oar/w .//
30
I L
(b)det(B102) 3 IL 9M3
: 9‘ = —? :9
'7}
(mama?) : M {as 8e) : My? .0410‘
: Jlié 9M8 (d) det(ZB)I : 9 V i M 8 I
<13
&
\
L.
1
ﬁg
\\ MATH 115, Midterm Page 4 of 11 Name: [9] 3. (5.} Write the system  _3 3K 2 —[ /
A, 4: gum 4,:(3) /
In. a ‘3 *. (b)FinclA".
.//
Iii—“lice 1.1! '00
/""/ 0(a) —7 OIO/[IO
’2000/ collot
4’
[.10 no:
“*7/0l0/1/0ﬂ loo‘ﬂ‘al
OO(,_/0/ 0/01/0
o0//o/
,, {'g‘ai /
A 5 [)0
via! (c) Use part (b) to solve the system. WA '1 o ’f 3' Page 5 of 11 MATH 115, Midterm
1 2 3 4
1 4 3 a
[7] 4.LetA= 12 0 3‘
o 2 0 3
(a) Find det(A).
// (b) Is A invertible? Explain. L/fJ .JH’IIca #0 / Name: MATH 115; Midterm Page 6 of 11 Name: 1 2 3
[8] 5. Write the matrix A = ( l 4 3 ) as a product of elementary matrices.
1 2 6 / .9 3 21"41'193 I 3 3
I V 3 h—z O .1 o
/ 3 C I 1 c I 00 Rt"y:.px job
_ OJ —
EW 30 E,)’DO 0/ o
'1’ 0:0) 0 ° 5’ 01¢ 00.}
06" 0o!
Raﬁ/21.6) I Do
7 0/0 =3:
1’00 00/
_¢:/O/0
0°73. MATH 115, Midterm Page 7 of 11 Name: [6] 5. Two vectors x and y are called orthogonal if x y = 0. Find all vectors x = (.131, 2:2, 2:3, 3:4) that
are orthogonal to both (1,2, —1,2) and (2, 3, —5, 3). “oi(I/Q—r,1} :. ac, +3.0LL9(3+Q.9H1 I //
si (3,3,s,3) : 0.}, +32% 4x1. 7‘33“?
5° U6. [14“.
at, 491., ~13+13‘Lr=° /
to}, 4—31.. we‘ll. 1‘3"!”
(la’1/°) ? l_3—l.lo
// MATH 115, Midterm Page 8 of 11 Name: 1+5 $27 553
[7] 7. LetAm ( $1 1+x2 1:3 Show that det(A) =1+I1+$2+I3.
$1 $2 1+$3
gig—hi ((a/IJMA apo)
 IN», 41.. Hi, 31. 7‘; 4M "I"; f”1 +11 IJDl‘. 3.1 (J a “MHzl is ax In, 7" “4/
: ﬂint, +1.11%) 1 31., at:
I lfJh at;
I UL (Ma
: (Ll—.35 #11 +11)/ I 0 a 5kg, atrcaf/ .Bm‘a/z
I l o _
g o / 424.6 3. ccf/ Freud} MATH 115, Midterm ‘ Page 9 of 11 Name: [8] 8. Find conditions on (1,5 and c such that the system 3:1 + 1122 +3z3 = a
:17: +2.73 ﬂ b
2x1 + $2 +5223 = c is consistent. I I 3 an , , 3 .q
0 I I A —) 6, , A
.2 I 5 c. o,,_' (“a
——n 4:: 5 xx
° ° 0 44:46 ‘ MATH 115, Midterm Page 10 of 11 Name: [9] 9. Determine if the following statements are true or false. Provide a brief justiﬁcation for your answer.
K/
(e) If A2 = A, then A is invertible. F. u Aw" 44.. AE/l M "AM
/ a, (b) Suppose Ax = b is on m x 7:. system with m < n. Then‘ the system must have inﬁnitely
many solutions. F 17% SUuFM'W‘O 1mm 13." alum w / (c) Suppose X; and x2 are both solutions to the homogenous system Ax = 0. Then the vector
y = 2x1 — 3):: is also a. solution to the homogeneous system. ./ 4, A33 A (99‘1"31") (I
= 9/15;  Man
2 :1. 3 — 33“ ;
=4 ...
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This note was uploaded on 05/18/2011 for the course MATH 115 taught by Professor Dunbar during the Spring '07 term at Waterloo.
 Spring '07
 DUNBAR
 Math, Linear Algebra, Algebra

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