JEE Main - Matrices and algebra of matrices Online Test (MCQ Quiz)

1. What is the order of a matrix if it has 3 rows and 4 columns?

A) 3x4 B) 4x3 C) 12 D) 3+4

2. Which of the following is NOT a type of matrix?

A) Row Matrix B) Column Matrix C) Square Matrix D) Circular Matrix

3. A matrix where all elements are zero is called a:

A) Identity Matrix B) Scalar Matrix C) Null Matrix D) Diagonal Matrix

4. 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.

5. 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]]

6. 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]]

7. 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)

8. 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

9. 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.

10. If A is a square matrix, then (A^T)^T is equal to:

A) A B) -A C) A^2 D) Null Matrix

11. A square matrix A is called symmetric if:

A) A = -A^T B) A = A^T C) A = A^2 D) A = Null Matrix

12. 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

13. For a skew-symmetric matrix, the diagonal elements are always:

A) 1 B) -1 C) 0 D) Any real number

14. 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

15. 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

16. 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]]

17. 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

18. What is the sum of elements of a matrix A = [[2, -1], [3, 5]]?

A) 5 B) 7 C) 9 D) 11

19. 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]]

20. 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

21. 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

22. 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]]

23. A matrix with only one row is called a:

A) Column Matrix B) Row Matrix C) Scalar Matrix D) Diagonal Matrix

24. A matrix with only one column is called a:

A) Row Matrix B) Column Matrix C) Square Matrix D) Null Matrix

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. 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]]

27. If A = [[a, b], [c, d]], what is the determinant of A?

A) ad + bc B) ac + bd C) ad - bc D) ab - cd

28. For a matrix A, if A = A^T, then A is:

A) Skew-Symmetric B) Symmetric C) Diagonal D) Orthogonal

29. For a matrix A, if A = -A^T, then A is:

A) Symmetric B) Skew-Symmetric C) Scalar D) Idempotent

30. 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

31. 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

32. What is the order of the product of a 2x3 matrix and a 3x2 matrix?

A) 2x2 B) 3x3 C) 2x3 D) 3x2

33. 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]]

34. 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]]

35. 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.

36. 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.

37. 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

38. 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

39. 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

40. 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

41. 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

42. 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

43. 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

44. The sum of elements on the main diagonal of a square matrix is called its:

A) Determinant B) Trace C) Order D) Rank

45. 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)

46. 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)

47. 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)

48. 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]]

49. 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

Matrices and algebra of matrices - Online Test

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