SET A LINEAR ALGEBRA

1. A linear system with only one solution is called

A. Consistent and Dependent C. Inconsistent and Dependent
B. Consistent and Independent D. Inconsistent

2. Which is true for the following system of equation?

3𝑥 + 5𝑦 = 8
9𝑥 + 15𝑦 = 24
A. System consists of non-intersecting lines.
B. System consists of overlapping lines.
C. System consists of lines intersecting only on one point.
D. System consists of parallel lines.

3. Who among the following is the proponent of matrix algebra?

A. Seki Kowa C. David Hilbert
B. Carl Friedrich Gauss D. Arthur Cayley

4. Who among the following mathematicians introduced a method in finding solutions to linear systems using
determinants?
A. Arthur Cayley C. Seki Kowa
B. Gabriel Cramer D. Augustin Louis Cauchy

5. Which of the following is FALSE?

A. An identity matrix is a scalar matrix. C. An identity matrix is a diagonal matrix.
B. A triangular matrix is a square matrix. D. A zero matrix is a square matrix.

−1 3 0
6. Identify the element 𝑎12 in 𝐴𝑇 of the given matrix 𝐴 = [ 2 1 4 ].
5 −3 −2
A. – 1 B. 2 C. 3 D. 12

7. The trace of a square matrix A is the sum of its diagonal elements. If

3 5 2 1 6 3
𝐴 = [2 4 5] , 𝐵 = [2 3 8]
6 7 1 7 1 9
Find the relationship between 𝑡𝑟(𝐴 + 𝐵) and 𝑡𝑟(𝐴) + 𝑡𝑟(𝐵).
A. tr(A+B)<tr(A)+tr(B) C. tr(A+B)≠tr(A)+tr(B)
B. tr(A+B)>tr(A)+tr(B) D. tr(A+B)=tr(A)+tr(B)

−2 1 4 2
8. Find 2𝐴 − 𝐵 if 𝐴 = [ ],𝐵 = [ ].
3 0 −5 7
0 0 0 −4 −8 −4 −8 0
A. [ ] B. [ ] C. [ ] D. [ ]
1 −7 11 −7 1 −7 11 −7

−7 6 −4 −1
9. Find the product [ ][ ].
1 6 −4 3
28 −24 4 11 4 25 4 11
A. [ ] B. [ ] C. [ ] D. [ ]
−7 18 −28 19 −28 17 19 −28

2 −8 4 −6
10. Find the matrix 𝑋 in the equation 𝑋 − 4 [ ]=[ ].
−4 2 2 8
−4 −38 12 26 12 −38 12 −38
A. [ ] B. [ ] C. [ ] D. [ ]
−14 −16 18 0 −14 16 −14 0

−14 −𝑤 2 2𝑘 −81
11. Find the values of the variables in the matrix equations [ ]=[ ].
3𝑓 3 3 3
A. 𝑓 = 1, 𝑘 = −7, 𝑤 = −9 𝑜𝑟 9 C. 𝑓 = −1, 𝑘 = −7, 𝑤 = −81 𝑜𝑟 81
B. 𝑓 = −1, 𝑘 = −7, 𝑤 = −9 𝑜𝑟 9 D. 𝑓 = 1, 𝑘 = −7, 𝑤 = 9

12. If matrix 𝐴 is 3 x 2, 𝐵 is 3 x 3, and 𝐶 is 2 x 3, which of the following operations CANNOT be performed?

A. 2A + 3C B. AB +C C. BA D. 3B + 5C.

1 3 −2
13. Find the inverse of the matrix [3 0 2 ]
1 4 −2
1 3 1 3 1 1
1 4 −4 1 4 −4 1 3 1 4
8 2
A. [−1 0 1 ] B. [2 −1 3 ] C. [ 3 0 4 ] D. [−1 0 1]
3 1 9 5 2
−5 8 8 9 −2 2 −2 −5 8 9
8 3
 −5 4
14. Solve for the inverse of the matrix 𝐴 = [ ].
0 4
1 1 1 1 1 1
−5 −5 −5 4 5
5
A. [ 1] B. [ 1] C. [ 1] D. Not Invertible
0 −4 0 0 −5
4
15. Which of the following is the determinant of a matrix obtained by deleting a row and a column?
A. Cofactor B. Minor C. Transpose D. Inverse
6 8
16. Determine whether the inverse of the matrix 𝐴 = [ ] exists.
3 4
A. Yes B. No C. sometimes exists D. cannot be determined
2 0 −2
17. Determine whether the inverse of the matrix 𝐵 = [1 3 4 ] exists.
2 6 8
A. Yes B. No C. sometimes exists D. cannot be determined
−10 −10
18. Compute the determinant of the matrix [ ].
5 −7
A. – 120 B. 20 C. 120 D. – 20

−3 1
19. Find the determinant of a given matrix 𝐴 = [ ]
−1 2
A. -3 B. -2 C. -5 D. -4
3 5 2
20. Solve for the determinant of the matrix [2 4 5].
6 7 1
A. 25 B. 26 C. 27 D. 28

21. Find 𝑑𝑒𝑡(𝐴𝑇 𝐵) if 𝑑𝑒𝑡(𝐴) = 3 and det(𝐵 𝑇 ) = 5.

A. – 15 B. 3/5 C. 5/3 D. 15

22. Find the area of the parallelogram with vertices (0, 0), (2, 6), (11,8), (7, 2) .
A. 42 B. 100 C. 50 D. 49

4 0 −1
23. Find the minor of the element 𝑎31 in the given matrix [5 2 −7].
2 3 4
A. 11 B. – 11 C. 2 D. – 2
4 0 −1
24. The cofactor of 2 in the given matrix [5 2 −7] is
1 3 4
A. -14 B. 14 C. – 18 D. 18
4 7
25. If 𝐴 = [ ], then 𝑎𝑑𝑗𝐴 is
−1 3
3 7 3 −7 3 −7 4 7
A. [ ] B. [ ] C. [ ] D. [ ]
1 3 1 4 2 3 −1 3

26. Find the y-value of the system

2𝑥 + 3𝑦 + 𝑧 = 2
𝑥 + 2𝑦 − 4 = −𝑧
3𝑥 + 𝑧 = −3𝑦 + 2
A. 0 B. -8 C. -2 D. 8
3 9
27. The complete row operations to find the inverse of the matrix 𝐶 = [ ]using Gauss-Jordan Elimination is
2 4
1 1 1 1
A. 3 𝑟1 , −2𝑟1 + 𝑟2 , − 2 𝑟2 , 3𝑟2 + 𝑟1 C. 3 𝑟1 , −2𝑟1 + 𝑟2 , − 2 𝑟2 , −3𝑟1 + 𝑟2
1 1 1 1
B. 𝑟 , 2𝑟1 + 𝑟2 , 2 𝑟2 , −3𝑟2 + 𝑟1 D. 3 𝑟1 , −2𝑟1 + 𝑟2 , − 2 𝑟2 , −3𝑟2 + 𝑟1
3 1

28. Which among the following is in Row Echelon Form?

0 1 2 0 0 1 0 2 0 0 1 1 2 0 0
1 0 0
0 0 0 1 0 0 1 0 0 0 0 1 0 1 0
A. [0 1 0] B. [ ] C. [ ] D. [ ]
0 0 0 0 1 0 0 0 1 0 0 0 1 0 1
0 1 1
0 0 0 0 1 0 0 0 1 0 0 0 0 0 0

2𝑥 − 𝑦 = −4
29. The linear system { has a(n)
6𝑥 − 3𝑦 = 12
A. unique solution B. infinite solution C. no solution D. unique and infinite solution

3𝑥 + 2𝑦 = 19
30. Solve for the value of 𝑦 in the linear system { .
𝑥 + 𝑦 = −8
A. -35 B. 35 C. 43 D. -43
31. Find the area of a triangle with the points (1,3), (4, 7), (2, − 3).
A. 11 sq units B. 12 sq units C. 13 sq units D. 22 sq units

32. The two vectors 𝑟 = 5𝑖 + 2𝑗 + 3𝑘 and 𝑟 = −2𝑖 + 5𝑘 are

A. Parallel B. Perpendicular C. mere intersecting D. Overlapping

33. Which of the following vectors with initial point at the origin is a translation of the vector with initial point at (-3, -2) and
a terminal point at (2, -1)?
A. [5 –3] B. [5 3] C. [–5 –3] D. [–5 3]

34. Find the direction number for the line that joins the initial point (2,4,1) and the terminal point (4, 4, 3).
The direction number is translated vector (with initial point at the origin) such that each component of the vector is
divided by the common factor of all the components.
A. [1 0 1] B. [1 0 -1] C. [-1 0 1] D. [-1 1 -1]

35. Which of the following is a magnitude of the vector with initial point at (-3, -2) and a terminal point at (-2, 1)?
A. 10 units B. 6.03 units C. 3.16 units D. 2.82 units

36. Which is an angle of the vector v = [4 − 3] from the positive x-axis, in a counterclockwise direction?
A. 59.04 degrees B. 120.96 degrees C. 233.13 degrees D. 53.13 degrees

37. Determine the angle from the positive 𝑥-axis (in counter-clockwise direction) and the magnitude of the vector [-2 3].
A. 123. 69° and 3.61 units C. 33.69° and 3.61 units
B. 132.69° and 6.31 units D. 176.31° and 6.31 units
1
38. Given that u = [1 7 2] and v = [2 6 - 10], perform the operation 3u − 2 v.
A. [-2 -18 -1] B. [3 21 6] C. [2 18 11] D. [2 18 1]

39. Given the vectors u=[5 2 3] and v=[– 2 –5 0], determine the cross product u x v.
A. [15 -6 -21] B. – 20 C. [–15 –6 29] D. [15 –6 21]

40. Given the vectors u=[5 2 3] and v=[–2 –5 0], determine the dot product u ∙ v.
A. 0 B. – 20 C. [–15 –6 29] D. [15 –6 21]

41. The angle between the vectors [1 − 2 3] and [2 4 5] is

A. 68.99° B. 78.99° C. 88.99° D. 98.99°

42. Given the vectors u=[3 -5 6] and v=[9 -15 12], perform the operation (u x v) ∙ u.
A. -90 B. 0 C. 90 D. 180

43. Given a position vector r = −3i + 2j − 10k. Which of the following is its corresponding vector in row matrix form?
A. [-3 -2 10] B. [3 2 10] C. [-3 2 -10] D. [-3 -2 -10]

44. Find the area of the parallelogram having the vectors u=3i – 5j + 6k and v=9i – 15j + 12k as the its adjacent sides.
A. 34.99 sq. units B. 45.18 sq. units C. 29.89 sq. units D. 37.88 sq. units

𝑎 𝑏
45. Given a set {[ ] ∈ 𝕄2𝑥2 | 𝑎 = 0}. Which is true about the given set?
𝑐 𝑑
A. The set is a vector space. C. The set is both a vector space and space.
B. The set is a subspace. D. The set is neither a vector space nor a subspace.

46. Given a set {𝕊 = {[𝑥 𝑦 𝑧] ∈ 𝑅 3 |𝑦 = 1}. Which among the following properties is not satisfied given that u, v, and w are
vectors while a and b are real numbers?
A. Zero vector must exist. C. u + (v + w) = (u + v) + w
B. u + v = v + u D. (𝑎 + 𝑏)u = 𝑎u + 𝑏u

47. All of the following are the main considerations to be satisfied for a vector space to be a subspace EXCEPT
A. The zero vector must exist. C. The set is closed under scalar multiplication
B. The set is closed under vector addition D. The vector addition must be commutative.

48. Which is TRUE about the set 𝑆 = {[𝑥 0 𝑥 + 1] ∈ 𝑅 3 | 𝑥 ∈ 𝑅}?

A. The set is a vector space only C. The set is both a vector space and a subspace
B. The set is a subspace only D. The set is neither a vector space nor a subspace

49. Which axiom below is not satisfied by the set 𝑆 = {[𝑥 0 2𝑥] ∈ 𝕄22 |𝑥, 𝑦 ∈ 𝑅}, under standard matrix addition and
scalar multiplication subspaces.
I. The zero vector must exist.
II. The set is closed under vector addition
III. The set is closed under scalar multiplication
A. I only B. I and II only C. II and III only D. none of the given choices
50. Who among the following mathematicians was the first one to use the word “eigen” in the word eigenvalues and
eigenvectors?
A. Leonhard Euler C. David Hilbert
B. Augustin-Louis Cauchy D. Joseph-Louis Lagrange

2 2
51. Given that 𝐴 = [ ], then the vector v = [2 − 5] is
5 −1
A. An eigenvector of A C. An eigenvalue of A
B. Not an eigenvector of A D. Cannot be determined

1 2
52. Verify whether the vector [1 2] is an eigen vector of [ ]. If not, determine the transformation.
0 1
A. Eigenvector C. Not an eigenvector, reflection
B. Not an eigenvector, shear D. Not an eigenvector, projection

4 2 4
53. If v = [3 1] is an eigenvector of [ ], find the eigenvalue.
3 1
A. 4/3 B. 2 C. 5 D. v is not an eigenvector

3 2
54. To find the eigenvalues of the matrix [ ], the characteristic equation is
3 8
2 2
A. 𝜆 + 11𝜆 − 18 = 0 B. 𝜆 − 11𝜆 + 18 = 0 C. 𝜆2 − 11𝜆 + 30 = 0 D. 𝜆2 − 11𝜆 − 30 = 0

3 2
55. One eigenvalue of [ ] is
3 8
A. 2 B. – 9 C. – 2 D. 6
−2 −4 2
56. If v = [2 − 3 − 1] is an eigenvector of [−2 1 2], then the eigenvalue corresponding to the eigenvector is
4 2 5
A. – 3 B. – 3 C. 3 D. 6
−6 3
57. The eigenvector of 𝐴 = [ ] corresponding to an eigenvalue of 6 can be expressed as
4 5
A. [4 1] B. [1 4] C. [1 − 4] D. [−4 1]

58. Solve for the unit vector of v = [12 16].

12 16 3 4 3 4 3 4
[25 ] B. [15 ] C. [20 ] D. [5 ]
25 15 20 5
3 −6
59. Given the vector 𝑣 = [3 2] and a matrix 𝐴 = [ ], which transformation describes the vector 𝐴𝑣?
2 2
A. Homothety B. Projection C. Shear D. Dilation

60. What is the significance of Eigenvalues and Eigenvectors in solving Linear Equations.
A. Eigenvalues and Eigenvectors do not have practical application
B. Eigenvalues and Eigenvectors is significant in describing Linear Transformation
C. Eigenvalues and Eigenvectors are only applicable in Polynomial Equations
D. Eigenvalues and Eigenvectors are used only in Mathematical Theories.

Draw the following 3D vectors.

61. v1 = [5 4 2]
62. v2 = [−2 3 0]
63. v3 = [0 3 − 1]
64. v4 = [−1 − 3 − 4]
65. v5 = [3 − 2 − 1]
---GOOD LUCK AND GOD BLESS---