MATH115MF05P

# MATH115MF05P - 501.0170 N3 MATH 115 Tuesday NAME Please...

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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 ISL-L: ——‘{ -5’t'r ‘ 0‘1 ‘1'“: '3’ L! ' SI: (‘13-) : (*V-Je z/M-l >4L/ 3-3. 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 .// 3-0 I L (b)det(B-102) 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/ col-lot 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, +3.0L-L-9(3+Q.9H1 I // si- (3,3,-s,3) : 0.}, +32% 4x1. 7‘33“? 5° U6. [14“. at, 4-91., -~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», 4-1.. Hi, 31. 7‘; 4M "I"; f”1 +11 IJ-Dl‘. 3.1 (J a “MHz-l is ax In, 7" “4/ : ﬂint, +1.11%) 1 31., at: I lf-Jh 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 — 3-3“ ;- =4 ...
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