JEE Main - Determinants of order two and three Online Test (MCQ Quiz)

1. What is the value of the determinant of a 2x2 matrix [[a, b], [c, d]]?

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

2. If A = [[2, 3], [4, 5]], what is the determinant of A, denoted as |A|?

A) 2 B) -2 C) 10 D) 23

3. What is the determinant of the identity matrix of order 2, I = [[1, 0], [0, 1]]?

A) 0 B) 1 C) 2 D) -1

4. For a 2x2 matrix A = [[p, q], [r, s]], if ps - qr = 0, what is the relationship between the rows or columns?

A) Rows are linearly independent B) Columns are linearly dependent C) Rows are proportional D) The matrix is non-singular

5. If A = [[a, b], [c, d]] is a 2x2 matrix, then det(kA) for a scalar k is equal to:

A) k * det(A) B) k^2 * det(A) C) det(A) D) k + det(A)

6. What is the value of the determinant of a 3x3 matrix [[a, b, c], [d, e, f], [g, h, i]]?

A) a(ei - fh) + b(fg - di) + c(dh - eg) B) a(ei - fh) - b(fg - di) + c(dh - eg) C) a(ei + fh) - b(fg + di) + c(dh + eg) D) a(ei - fh) + b(di - fg) + c(eg - dh)

7. If A = [[1, 2, 3], [0, 1, 4], [5, 6, 0]], what is the determinant of A?

A) 1 B) -19 C) 19 D) 0

8. What is the determinant of the identity matrix of order 3, I = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]?

A) 0 B) 3 C) 1 D) -1

9. If any two rows (or columns) of a 3x3 determinant are identical, what is the value of the determinant?

A) 1 B) -1 C) 0 D) The determinant is undefined

10. If a matrix A is multiplied by a scalar k, then det(kA) for a 3x3 matrix A is equal to:

A) k * det(A) B) k^3 * det(A) C) det(A) D) 3k * det(A)

11. What is the determinant of the matrix [[x, y], [y, x]]?

A) x^2 + y^2 B) x^2 - y^2 C) xy - yx D) 2xy

12. If A = [[a, b], [c, d]] and B = [[e, f], [g, h]], then det(AB) is equal to:

A) det(A) + det(B) B) det(A) * det(B) C) det(A) / det(B) D) det(B) / det(A)

13. What is the determinant of a matrix where one row is a scalar multiple of another row?

A) 1 B) -1 C) 0 D) The determinant is equal to the scalar multiple

14. If A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]], what is the determinant of A?

A) 1 B) -1 C) 0 D) 27

15. The determinant of a diagonal matrix of order 3, [[a, 0, 0], [0, b, 0], [0, 0, c]], is:

A) a + b + c B) abc C) 0 D) a^2 + b^2 + c^2

16. What is the determinant of the matrix [[5, 0], [0, 5]]?

A) 0 B) 5 C) 10 D) 25

17. If det(A) = 5 for a 2x2 matrix A, what is det(3A)?

A) 15 B) 45 C) 9 D) 125

18. What is the value of the determinant of the matrix [[1, 0, 0], [2, 1, 0], [3, 4, 1]]?

A) 0 B) 1 C) 6 D) 10

19. If A = [[a, b], [c, d]], then det(A^T) is equal to:

A) -det(A) B) det(A) C) 1/det(A) D) det(A)^2

20. What is the determinant of the matrix [[sin(theta), cos(theta)], [-cos(theta), sin(theta)]]?

A) 0 B) 1 C) -1 D) sin^2(theta) + cos^2(theta)

21. For a 3x3 matrix A, if det(A) = 0, what does this imply about the matrix A?

A) A is invertible B) A is non-singular C) A is singular D) The rows of A are linearly independent

22. If the elements of a determinant are all integers, what can be said about the value of the determinant?

A) It is always a rational number B) It is always an integer C) It is always an irrational number D) It can be any real number

23. What is the determinant of the matrix [[1, 2], [3, 4]]?

A) 10 B) -2 C) 2 D) 5

24. What is the determinant of the matrix [[a, b, c], [ka, kb, kc], [d, e, f]]?

A) k * det([[a, b, c], [a, b, c], [d, e, f]]) B) k * det([[a, b, c], [d, e, f], [d, e, f]]) C) k * det([[a, b, c], [d, e, f], [g, h, i]]) D) 0

25. If A = [[2, 1], [1, 2]], what is det(A^2)?

A) 3 B) 9 C) 81 D) 6

26. What is the determinant of the matrix [[1, 1, 1], [1, 2, 3], [1, 4, 9]]?

A) 0 B) 1 C) 2 D) 6

27. If A is a 2x2 matrix with det(A) = -3, what is det(-A)?

A) 3 B) -3 C) 9 D) -9

28. What is the determinant of the matrix [[a, b], [c, d]] if a, b, c, d are in arithmetic progression?

A) ad - bc B) 0 C) a+d D) b+c

29. If A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]], what is det(A) * det(B)?

A) -2 B) -24 C) 48 D) -4

30. What is the determinant of the matrix [[0, 0, 0], [1, 2, 3], [4, 5, 6]]?

A) 1 B) -1 C) 0 D) 12

31. If A is a 3x3 matrix and det(A) = 10, what is det(2A)?

A) 20 B) 40 C) 80 D) 100

32. What is the value of the determinant [[cos(theta), -sin(theta)], [sin(theta), cos(theta)]]?

A) 0 B) 1 C) -1 D) cos^2(theta) + sin^2(theta)

33. If A = [[1, 2, 3], [0, 0, 0], [4, 5, 6]], what is det(A)?

A) 1 B) 0 C) 3 D) 12

34. If A = [[a, b], [c, d]] is a 2x2 matrix and det(A) = 0, what can be said about the vectors [a, c] and [b, d]?

A) They are orthogonal B) They are linearly independent C) They are linearly dependent D) They are unit vectors

35. What is the determinant of the matrix [[1, 1, 1], [a, b, c], [a^2, b^2, c^2]] (Vandermonde determinant)?

A) (a-b)(b-c)(c-a) B) (b-a)(c-b)(a-c) C) (a+b+c) D) 0

36. If A = [[1, 2], [3, 4]] and B = [[0, 1], [1, 0]], what is det(AB)?

A) -2 B) 2 C) 0 D) -4

37. What is the determinant of the matrix [[1, 2, 3], [1, 2, 3], [4, 5, 6]]?

A) 0 B) 1 C) 6 D) 12

38. If det(A) = 5 and A is a 2x2 matrix, what is det(A^-1)?

A) 5 B) -5 C) 1/5 D) -1/5

39. What is the determinant of the matrix [[1, 0, 5], [0, 1, 6], [0, 0, 1]]?

A) 0 B) 1 C) 5 D) 6

40. If A = [[a, b], [c, d]] and det(A) = 0, then which of the following is true?

A) a/c = b/d B) a/b = c/d C) ad = bc D) ac = bd

41. What is the determinant of the matrix [[1, 2, 3], [4, 5, 6], [7, 8, 9]] after performing the operation R2 -> R2 + 2R1?

A) 0 B) 1 C) -1 D) 27

42. If A = [[2, 0], [0, 3]], what is det(A^3)?

A) 6 B) 18 C) 36 D) 216

43. What is the determinant of the matrix [[x, y, z], [x, y, z], [a, b, c]]?

A) xyz B) abc C) 0 D) x+y+z

44. If A is a 2x2 matrix with det(A) = 7, what is det(2A)?

A) 14 B) 28 C) 4 D) 49

45. What is the determinant of the matrix [[1, 0, 0], [0, 1, 0], [0, 0, 1]]?

A) 0 B) 1 C) 3 D) -1

46. If A = [[a, b], [c, d]], then det(A) = 0 implies:

A) a=0, b=0, c=0, d=0 B) a=b=c=d C) The rows (or columns) are linearly dependent D) The matrix is the identity matrix

47. What is the determinant of the matrix [[1, 2], [2, 4]]?

A) 0 B) 1 C) 4 D) 5

48. If det(A) = 3 for a 3x3 matrix A, what is det(A^T)?

A) 1/3 B) -3 C) 3 D) 9

49. What is the determinant of the matrix [[1, 0, 0], [2, 3, 0], [4, 5, 6]]?

A) 0 B) 1 C) 6 D) 18

50. If A = [[1, 2], [3, 4]] and B = [[1, 0], [0, 1]], what is det(A)?

A) -2 B) 2 C) 1 D) 5

Determinants of order two and three - Online Test

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