Matrices and algebra of matrices - Question Bank

1. If A is a matrix, then A * 0 (where 0 is the null matrix of appropriate order) is:
A) A
B) 0
C) I
D) 2A
2. If A = [[1, 2], [3, 4]] and B = [[-1, 0], [2, 1]], what is AB?
A) [[3, 2], [5, 4]]
B) [[1, 2], [3, 4]]
C) [[2, 2], [10, 6]]
D) [[3, 2], [10, 6]]
3. Which of the following is true for matrices A and B for which AB is defined?
A) Trace(AB) = Trace(BA)
B) Trace(AB) = Trace(A) * Trace(B)
C) Trace(AB) = Trace(A) + Trace(B)
D) Trace(AB) = Trace(A) - Trace(B)
4. If A is a square matrix and k is a scalar, then Trace(kA) is equal to:
A) k + Trace(A)
B) k * Trace(A)
C) Trace(A) / k
D) Trace(kA^2)
5. If A and B are matrices of the same order, then Trace(A + B) is equal to:
A) Trace(A) * Trace(B)
B) Trace(A) + Trace(B)
C) Trace(A) - Trace(B)
D) Trace(AB)
6. The sum of elements on the main diagonal of a square matrix is called its:
A) Determinant
B) Trace
C) Order
D) Rank
7. If A is a square matrix such that A^k = 0 (Null Matrix) for some positive integer k, then A is called:
A) Idempotent Matrix
B) Involutory Matrix
C) Nilpotent Matrix
D) Symmetric Matrix
8. If A is a square matrix such that A^2 = I (Identity Matrix), then A is called:
A) Idempotent Matrix
B) Nilpotent Matrix
C) Involutory Matrix
D) Scalar Matrix
9. If A is a square matrix such that A^2 = A, then A is called:
A) Nilpotent Matrix
B) Idempotent Matrix
C) Involutory Matrix
D) Periodic Matrix
10. If A is an m x n matrix and B is an n x m matrix, then AB and BA are:
A) Always of order m x m
B) Always of order n x n
C) Not necessarily of the same order
D) Always square matrices
11. Any square matrix can be expressed as the sum of:
A) Two symmetric matrices
B) Two skew-symmetric matrices
C) A symmetric and a skew-symmetric matrix
D) Two identity matrices
12. If A is a square matrix, A - A^T is always:
A) A symmetric matrix
B) A null matrix
C) A skew-symmetric matrix
D) An identity matrix
13. If A is a square matrix, A + A^T is always:
A) A skew-symmetric matrix
B) A null matrix
C) A symmetric matrix
D) A diagonal matrix
14. An identity matrix is a scalar matrix where:
A) All diagonal elements are zero.
B) All diagonal elements are equal but not necessarily one.
C) All diagonal elements are one.
D) All non-diagonal elements are one.
15. A scalar matrix is a diagonal matrix where:
A) All diagonal elements are different.
B) All diagonal elements are equal.
C) All diagonal elements are zero.
D) All diagonal elements are one.
16. If A = [[1, 2, 3]] and B = [[4], [5], [6]], what is AB?
A) [[4, 5, 6], [8, 10, 12], [12, 15, 18]]
B) [[1, 2, 3]]
C) [[32]]
D) [[4], [10], [18]]
17. If A = [[1], [2], [3]] and B = [[4, 5, 6]], what is AB?
A) [[4, 5, 6], [8, 10, 12], [12, 15, 18]]
B) [[1, 2, 3]]
C) [[4], [8], [12]]
D) [[1*4, 2*5, 3*6]]
18. What is the order of the product of a 2x3 matrix and a 3x2 matrix?
A) 2x2
B) 3x3
C) 2x3
D) 3x2
19. If A is an m x n matrix, then A^T is an:
A) m x n matrix
B) n x m matrix
C) m x m matrix
D) n x n matrix
20. Which of the following is a property of matrix addition?
A) Commutativity (A + B = B + A)
B) Associativity (A + (B + C) = (A + B) + C)
C) Existence of additive identity (A + 0 = A)
D) All of the above
21. For a matrix A, if A = -A^T, then A is:
A) Symmetric
B) Skew-Symmetric
C) Scalar
D) Idempotent
22. For a matrix A, if A = A^T, then A is:
A) Skew-Symmetric
B) Symmetric
C) Diagonal
D) Orthogonal
23. If A = [[a, b], [c, d]], what is the determinant of A?
A) ad + bc
B) ac + bd
C) ad - bc
D) ab - cd
24. If A = [[3, 2], [1, 4]] and B = [[1, -2], [-3, 5]], what is A - B?
A) [[2, 4], [4, -1]]
B) [[4, 0], [-2, 9]]
C) [[3, -2], [-1, 4]]
D) [[2, -4], [-4, 1]]
25. A matrix where the number of rows equals the number of columns is called a:
A) Row Matrix
B) Column Matrix
C) Square Matrix
D) Rectangular Matrix
26. A matrix with only one column is called a:
A) Row Matrix
B) Column Matrix
C) Square Matrix
D) Null Matrix
27. A matrix with only one row is called a:
A) Column Matrix
B) Row Matrix
C) Scalar Matrix
D) Diagonal Matrix
28. If A = [[1, 0], [0, 1]] and B = [[x, y], [z, w]], then AB is:
A) [[x, y], [z, w]]
B) [[1, 1], [1, 1]]
C) [[0, 0], [0, 0]]
D) [[xw, yz], [zx, wy]]
29. Which property states that for matrices A, B, and C (of appropriate orders), A(B + C) = AB + AC?
A) Commutative Property
B) Associative Property
C) Distributive Property
D) Identity Property
30. Which property states that for matrices A, B, and C (of appropriate orders), A(BC) = (AB)C?
A) Commutative Property
B) Associative Property
C) Distributive Property
D) Additive Inverse Property
31. If A = [[a, b], [c, d]], then -A is:
A) [[a, b], [c, d]]
B) [[-a, -b], [-c, -d]]
C) [[d, c], [b, a]]
D) [[a+b], [c+d]]
32. What is the sum of elements of a matrix A = [[2, -1], [3, 5]]?
A) 5
B) 7
C) 9
D) 11
33. If A is any matrix and I is the identity matrix of appropriate order, then AI is equal to:
A) Null Matrix
B) I
C) A
D) 2A
34. What is the identity matrix of order 2x2?
A) [[0, 0], [0, 0]]
B) [[1, 0], [0, 1]]
C) [[1, 1], [1, 1]]
D) [[1, 0], [1, 0]]
35. If A and B are matrices such that AB and BA are defined, then (AB)^T is equal to:
A) B^T * A^T
B) A^T * B^T
C) A^T + B^T
D) A^T - B^T
36. If A and B are matrices of the same order, then (A + B)^T is equal to:
A) A^T + B^T
B) A^T - B^T
C) A^T * B^T
D) B^T * A^T
37. For a skew-symmetric matrix, the diagonal elements are always:
A) 1
B) -1
C) 0
D) Any real number
38. A square matrix A is called skew-symmetric if:
A) A = A^T
B) A = -A^T
C) A = Null Matrix
D) A = Identity Matrix
39. A square matrix A is called symmetric if:
A) A = -A^T
B) A = A^T
C) A = A^2
D) A = Null Matrix
40. If A is a square matrix, then (A^T)^T is equal to:
A) A
B) -A
C) A^2
D) Null Matrix
41. What is the transpose of a matrix A, denoted by A^T?
A) A matrix obtained by interchanging rows and columns.
B) A matrix obtained by multiplying each element by -1.
C) A matrix obtained by adding all elements.
D) A matrix obtained by reversing the order of elements.
42. If A is an m x n matrix and B is an n x p matrix, what is the order of the product AB?
A) m x p
B) p x m
C) n x n
D) m x n
43. Matrix multiplication is:
A) Always commutative (AB = BA)
B) Never commutative (AB != BA)
C) Commutative only for square matrices
D) Not always commutative (AB may or may not be equal to BA)
44. If A = [[1, 2], [3, 4]], what is 2A?
A) [[2, 4], [6, 8]]
B) [[1, 4], [3, 8]]
C) [[3, 6], [9, 12]]
D) [[1, 2], [3, 4]]
45. If A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]], what is A + B?
A) [[6, 8], [10, 12]]
B) [[4, 4], [4, 4]]
C) [[5, 12], [21, 32]]
D) [[1, 2], [3, 4]]
46. What is the condition for two matrices to be equal?
A) They must have the same number of elements.
B) They must have the same order and corresponding elements must be equal.
C) They must be square matrices.
D) Their determinants must be equal.
47. A matrix where all elements are zero is called a:
A) Identity Matrix
B) Scalar Matrix
C) Null Matrix
D) Diagonal Matrix
48. Which of the following is NOT a type of matrix?
A) Row Matrix
B) Column Matrix
C) Square Matrix
D) Circular Matrix
49. What is the order of a matrix if it has 3 rows and 4 columns?
A) 3x4
B) 4x3
C) 12
D) 3+4
All Question Banks
E-Book Matrices and algebra of matrices One Line Questions Matrices and algebra of matrices Online Test Matrices and algebra of matrices