Lecturer: Phan Thi Khanh Van Approved by: Nguyen Tien Dung
UNIVERSITY OF TECHNOLOGY
- VNUHCM
FACULTY OF AS
MID. EXAM Semester/Academic year 3 21 - 22
Date 10/07/2022
Course title Linear Algebra
Course ID MT1007
Duration 50 mins Question sheet code 1007
Notes: - This is a closed book exam. Only your calculator is allowed. Total available score: 10.
- You MUST fill in your full name and student ID on this question sheet. There are 20 questions in 2 pages.
Question 1. Find all real values of msuch that the following linear system
3x1+x2−x3= 4
3x1+x2+mx3= 4
4x1+ 2x2+ 3x3= 5
has a unique
solution.
A.m6= 1 .B.m=−1.C.m6=−1.D.m= 1.E.m= 2.
Answer.
The system has a unique solution if and only if det(A)6= 0 ⇔ −2m−26= 0
The answer is C
Question 2. Find all real values of msuch that the vector (m, 1,3) belongs to the plane that is spanned by the
vector set M={(−1,2,1),(3,1,4)}.
A.m= 3.B.m= 2.C.m= 1.D.m= 0.E.m= 4.
Answer.
x∈span(M)if there exist α, β such that x=αe1+βe2
The answer is B
Question 3. Find all real values of msuch that the following linear system is consistent (has at least one solution):
x1+x2+x3= 1
x1+x2+mx3= 2 + m
3x1+ 3x2+ (m+ 2)x3=m2−2m
.
A.∀m∈R.B.m∈R\ {1,−1,4}.C.m=−1or m= 4 .
D.m∈R\ {−1,4}.E.m∈R\ {−1}.
Answer.
The answer is C
Question 4. Let A=Å1−1
−2 3 ãand B=Å1 5
−1−4ã. Find the matrix Xthat satisfies the equation
2AX =X+BT.
A.X=Å−1−17/3
−1−16/3ã.B.X=Å−1/3−1/3
−3−2ã.C.X=Å−5 13/3
−3 8/3ã.
D.X=Å−25/3−7/3
7 2 ã.E.X=Å3 13
−9−40ã.
Answer.
X= (2A−I)−1BT
The answer is C
Question 5. Let E, F and Gbe three bases of a two dimensional vector space V, with the change of basis matrices
TE→F=[e1]F[e2]F=Å1 2
−1 3ãand TG→F=[g1]F[g2]F=Å2 1
−1 2ã. Find the coordinate vector [x]Eof a
vector xwith respect to the basis Eif [x]G= (−2; 1)T.
A.(−17/5; 1/5)T.B.(−1; 2)T.C.(5; 15)T.D.(5; 10)T.E.(1; −2)T.
Answer.
[x]E=T−1
E→FTG→F[x]G
The answer is A
Question 6. Let Abe a 3×2matrix. Applying the elementary operation r2→r2+2r1is equivalent to multiplying
the following matrix
A.Å1 0
2 1ãto the right of A.B.Å1 0
2 1ãto the left of A.C.Ñ100
210
001éto the left of A.
Student ID. Number: ......................... Full name:.......................................... Page 1/4 – 1007
D.Å1 2
0 1ãto the right of A.E.Ñ120
010
001éto the left of A.
Answer.
The answer is C
Question 7. Find msuch that the dimension of the following nullspace: F=
x∈R4:
x1+ 2x2−x3+x4= 0
2x1+x2−x3+x4= 0
4x1+ 5x2−3x3+mx4= 0
is maximum.
A.m=−2.B.m= 2 .C.m= 4 .D.m= 3.E.m= 5.
Answer.
dim(F) = n−rank(A)is maximum if rank(A) is minimum, where A is the coefficient matrix.
The answer is D
Question 8. Let f(x) = x3−2x+ 3 and A=Å1 2
3 4ã. Find the determinant of f(A).
A.60.B.120.C.30.D.544.E.272.
Answer.
f(A) = A3−2A+ 3I
The answer is D
Question 9. Find msuch that det(A)=2where A=Ñ−123
0−4 5
0 0 1éÑ2m4
−222
1 1 4é.
A.m=9
20 .B.m=−6
5.C.m= 2.D.m=6
5.E.m=−3.
Answer.
|A|=|B|.|C|= 4.(10m−4)
The answer is A
Question 10. In a vector space V, let M={x, y, z}be a basis. Find the WRONG statement.
A.{2x, 3y, 4z}is not a spanning set of V.
B. M is linearly independent.
C.dim(V)=3.
D.N={3x−y−z, x +y+z, x +y}is linearly independent.
E.P={x, y, z, t}is linearly dependent, where t∈V.
Answer.
The answer is A
Question 11. Find all real values of msuch that M={(1; 1; −3),(m; 1; 2),(1; 1; 1),(2; 2; −2)}is a spanning set
of R3.
A.m6= 1.B.m6= 4.C.m6=−2.D.@m.E.∀m∈R.
Answer.
rank(M) = dim(R3)=3⇔m6= 1.
The answer is A
Question 12. Find all real values of msuch that A=Ñ1−1 2
3m2
−100éis invertible.
A.m= 3 .B.m6= 0.C.m6=−1.D.m6= 1.E.m= 1.
Answer.
A is invertible ⇔ |A| 6= 0
The answer is C
Question 13. Let Abe a 3×3matrix. If we apply the following operations: r1↔r2;r2→2r2;r3→r3+ 2r1,
we obtain the matrix Bwith the determinant det(B)=4. Find det(3A).
A.−6.B.−54.C.6.D.3.E.−27.
Answer.
|B|=−2|A|⇒|A|=−2.
|3A|= 33|A|=−54.
The answer is B
Student ID. Number: ......................... Full name:.......................................... Page 2/4 – 1007
Question 14. In R2, let E={(3; 1); (−2; −1)}be a basis and x= (4,5). Find [x]E.
A.(4,5)T.B.(17,−3)T.C.(−4,−5)T.
D. All other answers are wrong. E.(9,−23)T.
Answer.
The answer is D
Question 15-16:
For a simple economy with 3 industries A, B, C, we suggest the input-output model, where the input-output
matrix is A=Ñ0.3 0.2 0.3
0.1 0.15 0.05
0.1 0.2 0.1é. Given that the total productions of A, B and C are 500,400 and 300 units,
respectively.
Question 15. How many units of B that have been used to produce C?
A. 15. B. 60. C. 80. D. 20. E. 100.
Answer.
300 ∗0.05 = 15 (units)
The answer is A
Question 16. Find the external demand of A.
A.275.B.180.C.140.D.1157.E.642.
Answer.
D=X−AX
The answer is B
Question 17. In R4, let V=®(x1;x2;x3;x4)∈R4
®x1+x2+ 2x3+x4= 0
3x1+ 2x2+x3+ 2x4= 0 ´be a subspace. Find all real
values of mand nsuch that u= (3,2, m, n)∈V.
A.∀m, n.B.m= 1, n =−7.C.m=−23, n =−31.
D.m= 1, n = 2.E.@m, n.
Answer.
u∈V⇔uis a solution of the system.
The answer is B
Question 18. In an election there are two candidates A and B. At this week the percentages of the population
who support the candidate A and B are 51% and 49%, respectively. After each week, about 10% of the population
that support the candidate A change there mind and support the candidate B; 15% of population who support
B become A’s supporters. Estimate the percentage of the population who will support the candidate A after 1
month.
A.42.85%.B.57.15%.C.53.25%.D.46.75%.E.48%.
Answer.
Transition matrix: A=Å0.9 0.15
0.1 0.85ã.x0=Å0.51
0.49ã.
After 1 month: x4=A4x0=Å0.5715
0.4285ã.
The answer is B
Question 19. In the vector space P2[x] = {ax2+bx +c, a, b, c ∈R}, let M={x2+ 2x+ 1; 2x2+x;x+m}be a
vector set. Find all real values of msuch that Mis linearly dependent.
A.m=2
3.B.@m.C.m=1
3.D.m6=1
3.E.∀m∈R.
Answer.
M is linearly dependent if rank(M)<3⇔ |M|= 0 ⇔m=2
3.
The answer is A
Question 20. Find the current I1in the following electrical network, given that
R1= 1Ω, R2= 2Ω, R3= 3Ω, V1= 15V, V2= 12V.
A.57
11 .B.2.C.1
3.D.51
11 .E.5
9.
Answer.
Student ID. Number: ......................... Full name:.......................................... Page 3/4 – 1007
I1−I2−I3= 0
I1+ 2I2= 15
2I2−3I3= 12
The answer is D
Student ID. Number: ......................... Full name:.......................................... Page 4/4 – 1007
Lecturer: Phan Thi Khanh Van Approved by: Nguyen Tien Dung
UNIVERSITY OF TECHNOLOGY
- VNUHCM
FACULTY OF AS
MID. EXAM Semester/Academic year 3 21 - 22
Date 10/07/2022
Course title Linear Algebra
Course ID MT1007
Duration 50 mins Question sheet code 3622
Notes: - This is a closed book exam. Only your calculator is allowed. Total available score: 10.
- You MUST fill in your full name and student ID on this question sheet. There are 20 questions in 2 pages.
Question 1. Find msuch that det(A)=2where A=Ñ−123
0−4 5
0 0 1éÑ2m4
−222
1 1 4é.
A.m=−6
5.B.m=6
5.C.m= 2.D.m=9
20 .E.m=−3.
Answer.
The answer is D
Question 2. Find all real values of msuch that the following linear system is consistent (has at least one solution):
x1+x2+x3= 1
x1+x2+mx3= 2 + m
3x1+ 3x2+ (m+ 2)x3=m2−2m
.
A.∀m∈R.B.m∈R\ {1,−1,4}.C.m=−1or m= 4 .
D.m∈R\ {−1,4}.E.m∈R\ {−1}.
Answer.
The answer is C
Question 3. Find all real values of msuch that M={(1; 1; −3),(m; 1; 2),(1; 1; 1),(2; 2; −2)}is a spanning set of
R3.
A.m6=−2.B.m6= 1.C.@m.D.m6= 4.E.∀m∈R.
Answer.
The answer is B
Question 4. Let f(x) = x3−2x+ 3 and A=Å1 2
3 4ã. Find the determinant of f(A).
A.60.B.544.C.120.D.30.E.272.
Answer.
The answer is B
Question 5. Let A=Å1−1
−2 3 ãand B=Å1 5
−1−4ã. Find the matrix Xthat satisfies the equation
2AX =X+BT.
A.X=Å−25/3−7/3
7 2 ã.B.X=Å−1/3−1/3
−3−2ã.C.X=Å−1−17/3
−1−16/3ã.
D.X=Å−5 13/3
−3 8/3ã.E.X=Å3 13
−9−40ã.
Answer.
The answer is D
Question 6. In R2, let E={(3; 1); (−2; −1)}be a basis and x= (4,5). Find [x]E.
A.(4,5)T.B.(17,−3)T.C.(−4,−5)T.
D. All other answers are wrong. E.(9,−23)T.
Answer.
The answer is D
Question 7. In a vector space V, let M={x, y, z}be a basis. Find the WRONG statement.
A. M is linearly independent.
B.N={3x−y−z, x +y+z, x +y}is linearly independent.
C.dim(V)=3.
D.{2x, 3y, 4z}is not a spanning set of V.
E.P={x, y, z, t}is linearly dependent, where t∈V.
Answer.
Student ID. Number: ......................... Full name:.......................................... Page 1/3 – 3622