LINEAR ALGEBRA MCQ’S
Linear Equations..........................................................................................................................2
Matrices........................................................................................................................................ 5
Homogeneous and Non-Homogeneous Systems.................................................................. 13
Gaussian Elimination Method.................................................................................................. 20
Computing Determinants & Higher Order Determinants....................................................... 26
Determinant................................................................................................................................28
Permutation................................................................................................................................31
Expansion of Matrices...............................................................................................................32
Prepared By: Abdul Rafay​
Converted By: Muhammad Waleed
===========================================================
Linear Equations
===========================================================
1. Which of the following is a linear equation in one variable?
●​ A. x² + 2x = 0
●​ B. 2x + 3 = 7
●​ C. 1/x = 4
●​ D. x³ - 1 = 0
Answer: B
2. Solve: 3x + 5 = 20
●​ A. 5
●​ B. 10
●​ C. 15
●​ D. 20
Answer: A
3. What is the solution of 5x = 25?
●​ A. 10
●​ B. 0
●​ C. 5
●​ D. 1
Answer: C
4. The graph of a linear equation in two variables is a:
●​ A. Curve
●​ B. Circle
●​ C. Straight Line
●​ D. Parabola
Answer: C
5. Solve: 4x − 7 = 9
●​ A. 2
●​ B. 3
●​ C. 4
●​ D. 5
Answer: C
6. If 2x + 3 = 2x + 5, then
●​ A. x = 1
●​ B. x = 0
●​ C. No solution
●​ D. Infinite solutions
Answer: C
7. Find the value of x: x/3 + 2 = 5
●​ A. 6
●​ B. 9
●​ C. 1
●​ D. 3
Answer: B
8. The equation x + y = 10 represents a:
●​ A. Point
●​ B. Line
●​ C. Circle
●​ D. Parabola
Answer: B
9. Solve: 7x = 0
●​ A. x = 0
●​ B. x = 1
●​ C. x = 7
●​ D. No solution
Answer: A
10. Which of the following is not a linear equation?
●​ A. x + 2 = 5
●​ B. x² + 1 = 0
●​ C. 4x - 3 = 1
●​ D. 2x = 6
Answer: B
===========================================================
Matrices
===========================================================
11. What is the order of a matrix with 3 rows and 4 columns?
●​ A. 3 × 3
●​ B. 4 × 3
●​ C. 3 × 4
●​ D. 4 × 4
Answer: C
12. The matrix with first row [2, 3], second row [4, 5] is a:
●​ A. Row matrix
●​ B. Column matrix
●​ C. Square matrix
●​ D. Diagonal matrix
Answer: C
13. A matrix with only one row is called:
●​ A. Column matrix
●​ B. Row matrix
●​ C. Scalar matrix
●​ D. Identity matrix
Answer: B
14. The transpose of a matrix is obtained by:
●​ A. Squaring each element
●​ B. Reversing the rows
●​ C. Swapping rows with columns
●​ D. Replacing each element with its negative
Answer: C
15. Which matrix has all diagonal elements equal and all
off-diagonal elements zero?
●​ A. Identity matrix
●​ B. Scalar matrix
●​ C. Diagonal matrix
●​ D. All of the above
Answer: D
16. The product of a matrix and its inverse is:
●​ A. Zero matrix
●​ B. Diagonal matrix
●​ C. Identity matrix
●​ D. Transpose matrix
Answer: C
17. A matrix is symmetric if:
●​ A. A = A⁻¹
●​ B. A = Aᵀ
●​ C. A = −A
●​ D. A = A²
Answer: B
18. What is the determinant of the matrix with first row [1, 2],
second row [3, 4]?
●​ A. -2
●​ B. 10
●​ C. -5
●​ D. 2
Answer: A
19. Which of the following is not a type of matrix?
●​ A. Diagonal matrix
●​ B. Horizontal matrix
●​ C. Scalar matrix
●​ D. Identity matrix
Answer: B
20. If A is a 2×3 matrix and B is a 3×2 matrix, then the product
AB is of order:
●​ A. 3 × 3
●​ B. 2 × 2
●​ C. 3 × 2
●​ D. 2 × 3
Answer: B
21. The rank of a matrix is defined as:
●​ A. Number of non-zero rows
●​ B. Order of the matrix
●​ C. Number of rows
●​ D. Highest order of a non-zero minor
Answer: D
22. A zero matrix is a matrix:
●​ A. With all elements 1
●​ B. With only diagonal elements non-zero
●​ C. With all elements 0
●​ D. With elements equal to the order
Answer: C
23. What is the identity matrix of order 2?
●​ A. [[1, 0], [0, 1]]
●​ B. [[0, 0], [0, 0]]
●​ C. [[1, 1], [1, 1]]
●​ D. [[2, 0], [0, 2]]
Answer: A
24. The trace of a matrix is:
●​ A. Sum of diagonal elements
●​ B. Product of diagonal elements
●​ C. Sum of all elements
●​ D. Determinant of the matrix
Answer: A
25. A matrix A is said to be singular if:
●​ A. det(A) = 1
●​ B. det(A) = 0
●​ C. A = Aᵀ
●​ D. A = −A
Answer: B
26. Which operation is not defined for matrices of different
orders?
●​ A. Multiplication
●​ B. Addition
●​ C. Subtraction
●​ D. Both B and C
Answer: D
27. The inverse of a matrix exists only if:
●​ A. It is symmetric
●​ B. It is singular
●​ C. It is square and non-singular
●​ D. It has equal row and column sums
Answer: C
28. A matrix A such that A² = A is called:
●​ A. Identity matrix
●​ B. Involutory matrix
●​ C. Idempotent matrix
●​ D. Orthogonal matrix
Answer: C
29. An orthogonal matrix A satisfies:
●​ A. Aᵀ = A
●​ B. A⁻¹ = Aᵀ
●​ C. A = −A
●​ D. A² = I
Answer: B
30. Which of the following is a square matrix?
●​ A. 3x2
●​ B. 2x3
●​ C. 4x4
●​ D. 1x2
Answer: C
31. What is the order of the matrix with 3 rows and 2 columns?
●​ A. 2x3
●​ B. 3x2
●​ C. 6x1
●​ D. 1x6
Answer: B
32. Which matrix has only one row?
●​ A. Column matrix
●​ B. Square matrix
●​ C. Row matrix
●​ D. Null matrix
Answer: C
33. The determinant of a 2x2 matrix with first row [a, b], second
row [c, d] is:
●​ A. ab + cd
●​ B. ad – bc
●​ C. ac – bd
●​ D. a + d
Answer: B
34. A matrix with all elements 0 is called:
●​ A. Identity matrix
●​ B. Scalar matrix
●​ C. Zero matrix
●​ D. Diagonal matrix
Answer: C
35. Which of the following matrices is symmetric?
●​ A. [[1, 2], [3, 4]]
●​ B. [[1, 0], [0, 1]]
●​ C. [[0, 1], [1, 0]]
●​ D. [[1, 2], [2, 1]]
Answer: D
36. The transpose of a matrix A is denoted by:
●​ A. A²
●​ B. Aᵀ
●​ C. T(A)
●​ D. A’
Answer: B
===========================================================
Homogeneous and Non-Homogeneous Systems
===========================================================
37. A system of linear equations is called homogeneous if:
●​ A. It has at least one solution
●​ B. All the constants on the right-hand side are zero
●​ C. It has more equations than variables
●​ D. It has no solution
Answer: B
38. Which of the following is a homogeneous system?
●​ A. x + y = 2
●​ B. x + y = 0
●​ C. x – y = 5
●​ D. x + 2y = 1
Answer: B
39. The trivial solution of a homogeneous system is:
●​ A. The solution where all variables are 1
●​ B. The solution where all variables are 0
●​ C. A solution where one variable is 0
●​ D. No solution
Answer: B
40. A homogeneous system always has:
●​ A. No solution
●​ B. A unique non-trivial solution
●​ C. A trivial solution
●​ D. An infinite number of solutions
Answer: C
41. A non-homogeneous system:
●​ A. Always has a trivial solution
●​ B. Always has a non-trivial solution
●​ C. May or may not have a solution
●​ D. Has only complex solutions
Answer: C
42. Which of the following is a non-homogeneous system?
●​ A. x – y = 0
●​ B. 2x + 3y = 0
●​ C. x + y = 4
●​ D. x – y = 0 and 2x + 3y = 0
Answer: C
43. A homogeneous system of equations with more variables
than equations:
●​ A. Always has only the trivial solution
●​ B. Has no solution
●​ C. Has infinitely many solutions
●​ D. Is inconsistent
Answer: C
44. The system of equations: x + y = 0, 2x + 2y = 0 is:
●​ A. Homogeneous and consistent
●​ B. Non-homogeneous and inconsistent
●​ C. Homogeneous and inconsistent
●​ D. Non-homogeneous and consistent
Answer: A
45. Which of the following is always consistent?
●​ A. A non-homogeneous system
●​ B. A homogeneous system
●​ C. An overdetermined system
●​ D. A system with more equations than variables
Answer: B
46. The solution set of a homogeneous system of equations
forms:
●​ A. A single point
●​ B. A vector space
●​ C. A circle
●​ D. A plane
Answer: B
47. In a homogeneous system, if a non-trivial solution exists, the
system has:
●​ A. No solution
●​ B. Only the zero solution
●​ C. Infinitely many solutions
●​ D. A unique non-zero solution
Answer: C
48. A non-homogeneous system can be inconsistent. This means:
●​ A. It has a unique solution
●​ B. It has infinitely many solutions
●​ C. It has no solution
●​ D. It is always solvable
Answer: C
49. Which of the following systems is inconsistent?
●​ A. x + y = 1, x – y = 0
●​ B. x + y = 2, x + y = 2
●​ C. x + y = 1, x + y = 3
●​ D. x – y = 0, y – x = 0
Answer: C
50. The rank of the coefficient matrix of a homogeneous system
is less than the number of unknowns. What does this imply?
●​ A. No solution
●​ B. Unique solution
●​ C. Trivial and non-trivial solutions
●​ D. The system is inconsistent
Answer: C
51. For the system Ax = 0, if det(A) ≠ 0, then the solution is:
●​ A. No solution
●​ B. Infinite solutions
●​ C. Only trivial solution
●​ D. Non-trivial solution
Answer: C
52. The system: x – y + z = 0, 2x – 2y + 2z = 0 is:
●​ A. Non-homogeneous and consistent
●​ B. Homogeneous with only the trivial solution
●​ C. Homogeneous with infinite solutions
●​ D. Inconsistent
Answer: C
53. Which of the following matrices corresponds to a
homogeneous system?
●​ A. [1 2 | 3]
●​ B. [1 2 | 0]
●​ C. [0 0 | 1]
●​ D. [2 3 | 4]
Answer: B
54. A system of the form Ax = b, where b ≠ 0, is:
●​ A. Always homogeneous
●​ B. Always has a trivial solution
●​ C. Non-homogeneous
●​ D. Inconsistent
Answer: C
55. Which of the following is not true about homogeneous
systems?
●​ A. They are always consistent
●​ B. They may have infinite solutions
●​ C. They may have no solution
●​ D. They always have the trivial solution
Answer: C
56. The number of solutions of a homogeneous system of 3
equations in 4 unknowns is:
●​ A. 0
●​ B. 1
●​ C. Infinite
●​ D. 3
Answer: C
===========================================================
Gaussian Elimination Method
===========================================================
57. What is the primary purpose of Gaussian Elimination?
●​ A. To find the determinant of a matrix
●​ B. To solve systems of linear equations
●​ C. To find the inverse of a matrix
●​ D. To solve nonlinear equations
Answer: B
58. Which of the following is a key step in Gaussian Elimination?
●​ A. Finding the determinant
●​ B. Performing row operations
●​ C. Finding the inverse
●​ D. Solving nonlinear equations
Answer: B
59. What is the result of applying Gaussian Elimination to a
matrix?
●​ A. Upper triangular matrix
●​ B. Lower triangular matrix
●​ C. Diagonal matrix
●​ D. Identity matrix
Answer: A
60. Which row operation is used to eliminate variables in
Gaussian Elimination?
●​ A. Row addition
●​ B. Row multiplication
●​ C. Row swapping
●​ D. All of the above
Answer: D
61. What is the purpose of pivoting in Gaussian Elimination?
●​ A. To avoid division by zero
●​ B. To reduce round-off errors
●​ C. To improve numerical stability
●​ D. All of the above
Answer: D
62. Gaussian Elimination can be used to solve systems of linear
equations with:
●​ A. Unique solutions
●​ B. No solutions
●​ C. Infinite Solutions
●​ D. All of the Above
Answer: D
63. What is the time complexity of Gaussian Elimination?
●​ A. O(n²)
●​ B. O(n³)
●​ C. O(n⁴)
●​ D. O(n⁵)
Answer: B
64. Gaussian Elimination is also known as:
●​ A. Row reduction
●​ B. Column reduction
●​ C. Matrix inversion
●​ D. Determinant calculation
Answer: A
65. Which of the following matrices can be solved using Gaussian
Elimination?
●​ A. Singular matrix
●​ B. Non-singular matrix
●​ C. Both a and b
●​ D. Neither a nor b
Answer: C
66. What is the result of applying Gaussian Elimination to a
singular matrix?
●​ A. Upper triangular matrix
●​ B. Lower triangular matrix
●​ C. Inconsistent system
●​ D. Dependent system
Answer: D
67. Gaussian Elimination is named after:
●​ A. Carl Friedrich
●​ Isaac Newton
●​ Albert Einstein
●​ Leonhard Euler
Answer: A
===========================================================
Gauss-Jordan Method
===========================================================
68. What is the primary purpose of the Gauss-Jordan Method?
●​ A. To find the determinant of a matrix
●​ B. To solve systems of linear equations
●​ C. To find the inverse of a matrix
●​ D. To solve nonlinear equations
Answer: B
69. Which of the following is a key step in the Gauss-Jordan
Method?
●​ A. Finding the determinant
●​ B. Performing row operations
●​ C. Finding the inverse
●​ D. Solving nonlinear equations
Answer: B
70. What is the result of applying the Gauss-Jordan Method to a
matrix?
●​ A. Upper triangular matrix
●​ B. Lower triangular matrix
●​ C. Reduced row echelon form (RREF)
●​ D. Identity matrix
Answer: C
71. Which row operation is used to eliminate variables in the
Gauss-Jordan Method?
●​ A. Row addition
●​ B. Row multiplication
●​ C. Row swapping
●​ D. All of the above
Answer: D
72. What is the purpose of pivoting in the Gauss-Jordan Method?
●​ A. To avoid division by zero
●​ B. To reduce round-off errors
●​ C. To improve numerical stability
●​ D. All of the above
Answer: D
73. What is the time complexity of the Gauss-Jordan Method?
●​ A. O(n²)
●​ B. O(n³)
●​ C. O(n⁴)
●​ D. O(n⁵)
Answer: B
74. The Gauss-Jordan Method is also known as:
●​ A. Row reduction
●​ B. Column reduction
●​ C. Matrix inversion
●​ D. Gaussian elimination
Answer: A
===========================================================
Computing Determinants & Higher Order
Determinants
===========================================================
75. What is the determinant of a 2x2 matrix A = [[a, b], [c, d]]?
●​ A. ad – bc
●​ B. ad + bc
●​ C. ab – cd
●​ D. ab + cd
Answer: A
76. Which of the following properties is true for determinants?
●​ A. det(A + B) = det(A) + det(B)
●​ B. det(AB) = det(A)det(B)
●​ C. det(A) = det(Aᵀ)
●​ D. Both B and C
Answer: D
77. What is the determinant of a 3x3 matrix A = [[1, 2, 3], [4, 5, 6],
[7, 8, 9]]?
●​ A. 0
●​ B. 1
●​ C. -1
●​ D. 2
Answer: A
78. How does the determinant of a matrix change when two rows
are interchanged?
●​ A. It remains the same
●​ B. It becomes zero
●​ C. It changes sign
●​ D. It doubles
Answer: C
79. What is the determinant of a matrix that has two identical
rows?
●​ A. 1
●​ B. -1
●​ C. 0
●​ D. Cannot be determined
Answer: C
===========================================================
Determinant
===========================================================
80. If A is a square matrix, what is the relationship between
det(A) and det(Aᵀ)?
●​ A. det(A) = -det(Aᵀ)
●​ B. det(A) = det(Aᵀ)
●​ C. det(A) = 1/det(Aᵀ)
●​ D. det(A) = 0
Answer: B
81. If A and B are two square matrices of the same size, what is
the determinant of AB?
●​ A. det(A) + det(B)
●​ B. det(A) - det(B)
●​ C. det(A)det(B)
●​ D. det(A) / det(B)
Answer: C
82. If A is a square matrix and k is a scalar, what is the
determinant of kA?
●​ A. k det(A)
●​ B. k² det(A)
●​ C. kⁿ det(A), where n is the matrix order
●​ D. det(A)
Answer: C
83. If A is a singular matrix, what is the value of det(A)?
●​ A. 1
●​ B. -1
●​ C. 0
●​ D. Cannot be determined
Answer: C
84. If two rows or columns of a matrix are identical, what is the
determinant of the matrix?
●​ A. 1
●​ B. -1
●​ C. 0
●​ D. Cannot be determined
Answer: C
85. If A is an invertible matrix, what is the determinant of A⁻¹?
●​ A. det(A)
●​ B. 1/det(A)
●​ C. -det(A)
●​ D. 0
Answer: B
86. If two rows or columns of a matrix are interchanged, what
happens to the determinant?
●​ A. It remains the same
●​ B. It changes sign
●​ C. It doubles
●​ D. It becomes zero
Answer: B
87. If A is a square matrix and det(A) = 0, what can be concluded
about A?
●​ A. A is invertible
●​ B. A is not invertible
●​ C. A is symmetric
●​ D. A is skew-symmetric
Answer: B
===========================================================
Permutation
===========================================================
88. What is the effect on the determinant of a matrix when two
rows are interchanged?
●​ A. The determinant remains the same
●​ B. The determinant changes sign
●​ C. The determinant doubles
●​ D. The determinant becomes zero
Answer: B
89. If a matrix A has a determinant of 5, what is the determinant
of the matrix obtained by interchanging two rows of A?
●​ A. 5
●​ B. -5
●​ C. 10
●​ D. 0
Answer: B
90. What is the effect on the determinant of a matrix when a row
is permuted to a different position?
●​ A. The determinant remains the same
●​ B. The determinant changes sign for each permutation
●​ C. The determinant doubles for each permutation
●​ D. The determinant becomes zero
Answer: B
91. If P is a permutation matrix, what is the determinant of P?
●​ A. 0
●​ B. 1
●​ C. -1
●​ D. Depends on the specific permutation
Answer: D
92. What is the property of the determinant of a matrix product
PA, where P is a permutation matrix and A is any square matrix?
●​ A. det(PA) = det(A)
●​ B. det(PA) = -det(A)
●​ C. det(PA) = det(P)det(A)
●​ D. det(PA) = det(A)det(P)²
Answer: C
===========================================================
Expansion of Matrices
===========================================================
93. What is the expansion of a 2x2 matrix A = [[a, b], [c, d]]
along the first row?
●​ A. a(d) - b(c)
●​ B. a(d) + b(c)
●​ C. a(d) - c(b)
●​ D. ad - bc
Answer: C
94. What is the definition of the expansion of a matrix along a
row or column?
●​ A. The sum of the products of the elements of the row or column
and their cofactors
●​ B. The product of the elements of the row or column
●​ C. The sum of the elements of the row or column
●​ D. The difference of the elements of the row or column
Answer: A
95. What is the expansion of a 3x3 matrix A = [[a, b, c], [d, e, f],
[g, h, i]] along the first row?
●​ A. a(ei - fh) - b(di - fg) + c(dh - eg)
●​ B. a(ei + fh) - b(di + fg) + c(dh + eg)
●​ C. a(ei - fh) + b(di - fg) + c(dh - eg)
●​ D. a(ei + fh) + b(di + fg) + c(dh + eg)
Answer: A
96. What is the purpose of expanding a matrix along a row or
column?
●​ A. To find the determinant of the matrix
●​ B. To find the inverse of the matrix
●​ C. To solve a system of linear equations
●​ D. To find the rank of the matrix
Answer: A
97. What is the cofactor expansion of a matrix?
●​ A. The sum of the products of the elements of a row or column
and their minors
●​ B. The sum of the products of the elements of a row or column
and their cofactors
●​ C. The product of the elements of a row or column and their
minors
●​ D. The difference of the elements of a row or column and their
cofactors
Answer: B
98. For which of the following matrices is the determinant equal
to zero?
●​ A. A diagonal matrix with non-zero diagonal elements
●​ B. A matrix with two identical rows
●​ C. An upper triangular matrix with non-zero diagonal elements
●​ D. A lower triangular matrix with non-zero diagonal elements
Answer: B
99. If matrix A satisfies A² = A, then (I − A)³ + A equals:
●​ A. I
●​ B. 0
●​ C. I − A
●​ D. I + A
Answer: A
100. What is the cofactor of element a in a 2x2 matrix A = [[a, b],
[c, d]] when expanding along the first row?
●​ A. d
●​ B. -d
●​ C. c
●​ D. -c
Answer: A