Mathematics 1229A/B Study Guide - Midterm Guide: Augmented Matrix, Gaussian Elimination, Invertible Matrix
6 Jan 2019
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Friday, Novemb er 13 , 2 0 1 5
Page 1 CODE 111 Mathematics 1229A
Test 2
PART A (18 marks)
NOTE: YOUR ANSWERS TO THE PROBLEMS IN PART A MUST BE
INDICATED ON THE SCANTRON SHEET. YOU SHOULD ALSO CIRCLE
YOUR ANSWERS IN THIS BOOKLET.
1.1
mark Which of the following equations is/are linear in the unknowns x,yand z?
(i)x
3+3
y+z
3=6 (ii)x(1 + y+z)=3 (iii)√5x+5
2y+z
5=11
A:(i)and(ii)only B:(i)and(iii)only C:(ii)and(iii)only
D:(iii)only E:noneof(i), (ii)or(iii)
2.1
mark Find the augmented matrix corresponding to the following system of linear equations:
x=y+1
y=−z
A:!11 1 0
01−1 0 "B:!1−110
01−1 0 "C:!1−101
0110"
D:!1101
0110"E:!1−101
01−1 0 "
3.1
mark Which of the following matrices is/are in row-reduced echelon form?
(i)#010
001
$(ii)#120
001
$(iii)#102
012
$
(iv)⎡
⎢
⎣
001
010
100
⎤
⎥
⎦(v)⎡
⎢
⎣
100
000
001
⎤
⎥
⎦(vi)⎡
⎢
⎣
110
010
001
⎤
⎥
⎦
A:(i), (ii)and(iii)only B:(iv), (v)and(vi)only C:(i)and(vi)only
D:(v)and(vi)only E:NoneofA,B,CorD.
4.1
mark Find the solution to the system of linear equations whose augmented matrix is ⎡
⎢
⎣
222
011
000⎤
⎥
⎦.
A:(2,1,0) B:(1,1,0) C:(0,1) D:(1,0) E:Thereisnosolution.
5.1
mark Find all solutions to the system of linear equations whose augmented matrix is #1305
0010$.
A:Nosolutions. B:(5,0) only C:(5+3t, 0,t)D:(5−3t, 0,t)E:(5−3t, t, 0)
Mathematics 1229A
Test 2 CODE 111 Friday, November 13, 2015
Page 2
Use the following information for questions 6 and 7.
Consider the system of linear equations:
x−3y=1
kx +6y=1
6.1
mark For what value of kdoes the system have no solutions?
A:1 B:−1C:2 D:−2E:novalueofk
7.1
mark For what value of kdoes the system have infinitely many solutions?
A:1 B:−1C:2 D:−2E:novalueofk
8.1
mark Find all values of cfor which the solution to the system of linear equations corresponding
to the augmented matrix below has exactly one parameter.
⎡
⎢
⎣
102 3 4
011 2 3
000c2−1c+1 ⎤
⎥
⎦
A:c=0only B:c=1only C:c=−1only D:allc̸=±1E:allvaluesofc
9.1
mark The equations shown below represent planes in ℜ3.Describetheregion,ifany,thatisthe
intersection of these planes.
x−y+2z=1
−x+y−2z=−1
2x−y+5z=3
A:Exactlyonepoint. B:Exactlythreepoints. C:Aline.
D:Theplanesdonotintersect. E:NoneofA,B,CorD.
10.1
mark You are given that Ais a 2 ×3matrix,Bis a 3 ×2matrixandCis a 3 ×3matrix. Which
one of the following operations is defined?
A:A+C+BB:BA +C2C:CBATD:ABTE:AB −C
11.1
mark If 3 #1−2
−11
$−5#−1−1
34
$=#ab
cd
$,findthevalueofb.
A:−4B:−3C:−2D:−1E:NoneofA,B,CorD.
12.1
mark If A=#2−2
−11
$#−41
1−3$,findthe(1,2)-entry of A.
A:8 B:−4C:−2D:−10 E:5
60
MATH 1229A/B Full Course Notes
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Document Summary
Note: your answers to the problems in part a must be. Which of the following equations is/are linear in the unknowns x, y and z? (i) x. = 6 (ii) x(1 + y + z) = 3 (iii) 5 x + 52y + z. Find the augmented matrix corresponding to the following system of linear equations: 0 0 1 $ (i) # 0 1 0 (iv) . 0 0 1 $ (ii) # 1 2 0 (v) . 0 1 2 $ (iii) # 1 0 2 (vi) . Find the solution to the system of linear equations whose augmented matrix is . Find all solutions to the system of linear equations whose augmented matrix is# 1 3 0 5. C: (5 + 3t, 0, t) d: (5 3t, 0, t) Use the following information for questions 6 and 7. Consider the system of linear equations: x 3y = 1 kx + 6y = 1.
