JEE Main - Adjoint and inverse of a square matrix Question Bank with Options

1. If A is a non-singular matrix such that A adj(A) = [[1, 0, 0], [0, 1, 0], [0, 0, 1]], what is det(A)?

A) 1

B) -1

C) 0

D) Cannot be determined

2. If A is a non-singular matrix, then adj(kA) = kⁿ⁻¹ * adj(A), where n is the order of A and k is a non-zero scalar. This statement is:

A) True

B) False

C) True only if k=1

D) True only if n=2

3. Let A be a 3x3 matrix. If adj(A) = [[-1, 2, 0], [0, -3, 0], [0, 0, -4]], what is A⁻¹?

A) [[-1/12, 1/6, 0], [0, 1/4, 0], [0, 0, -1/3]]

B) [[1, -1/2, 0], [0, -1/3, 0], [0, 0, -1/4]]

C) [[-1, 0, 0], [2, -3, 0], [0, 0, -4]]

D) [[12, -6, 0], [0, -4, 0], [0, 0, 3]]

4. If A is a skew-symmetric matrix of even order, then det(A) is:

A) A perfect square

B) 0

C) 1

D) -1

5. If A is a skew-symmetric matrix of odd order, then det(A) is:

A) 0

B) 1

C) -1

D) Cannot be determined

6. For a non-singular matrix A, if adj(A) = A⁻¹, then det(A) must be:

A) 1 or -1

B) 1

C) -1

D) Any non-zero real number

7. Let A = [[1, 0], [0, 1]]. What is adj(A)?

A) [[1, 0], [0, 1]]

B) [[0, 0], [0, 0]]

C) [[1, 1], [1, 1]]

D) [[0, 1], [1, 0]]

8. If A is a 3x3 matrix and adj(A) = A, then A² is equal to:

A) I

B) A

C) -I

D) 2A

9. Let A be a 2x2 matrix such that A * adj(A) = [[5, 0], [0, 5]]. What is det(A)?

A) 5

B) -5

C) 25

D) √5

10. If A is a non-singular matrix, which of the following is NOT necessarily true?

A) adj(A) = det(A) * A⁻¹

B) A * adj(A) = det(A) * I

C) det(adj(A)) = (det(A))^(n-1)

D) adj(A) * A⁻¹ = I

11. If A is a non-singular matrix of order n, what is adj(A⁻¹)?

A) (adj(A))⁻¹

B) adj(A)

C) A

D) I

12. Let A be a 3x3 matrix. If adj(A) = [[1, 2, 3], [4, 5, 6], [7, 8, 9]], what is det(A)?

A) 0

B) 1

C) 27

D) 81

13. If A is a non-singular matrix, and adj(A) = [[2, 3], [4, 5]], and det(A) = -2, find A.

A) [[5/2, -3/2], [-2, -4]]

B) [[-5/2, 3/2], [2, 4]]

C) [[5, 4], [3, 2]]

D) [[2, 4], [3, 5]]

14. What is the adjoint of a diagonal matrix D = diag(d₁, d₂, ..., d<0xE2><0x82><0x99>)?

A) D itself.

B) diag(1/d₁, 1/d₂, ..., 1/d<0xE2><0x82><0x99>).

C) diag(d₁ⁿ⁻¹, d₂ⁿ⁻¹, ..., d<0xE2><0x82><0x99>ⁿ⁻¹).

D) The zero matrix.

15. If A = [[cos θ, sin θ], [-sin θ, cos θ]], what is A⁻¹?

A) [[cos θ, -sin θ], [sin θ, cos θ]]

B) [[cos θ, sin θ], [-sin θ, cos θ]]

C) [[-cos θ, sin θ], [sin θ, -cos θ]]

D) [[1/cos θ, 0], [0, 1/cos θ]]

16. Let A be a 3x3 matrix. If det(A) = 5, what is det(adj(A))?

A) 25

B) 125

C) 5

D) 1/5

17. If A is a non-singular matrix, then det(adj(A⁻¹)) is equal to:

A) (det(A))^(n-1)

B) det(A)

C) 1 / det(A)

D) (det(A))^(n-2)

18. What is the adjoint of a scalar matrix kI of order n?

A) k^(n-1) * I

B) k * I

C) I

D) n * k * I

19. If A is a non-singular matrix, when is adj(A) = A⁻¹?

A) When det(A) = 1.

B) When det(A) = -1.

C) When A = I.

D) When A² = I.

20. If A is a non-singular matrix, when is adj(A) = A?

A) When A² = I and det(A) = 1.

B) When A² = I.

C) When A = I.

D) When det(A) = 1.

21. If A is an n x n involutory matrix, what is its inverse?

A) A

B) I

C) Aᵀ

D) -A

22. If A is an n x n orthogonal matrix, what is its inverse?

A) Aᵀ

B) A

C) I

D) -A

23. Which of the following statements is always true for a non-singular matrix A?

A) A * adj(A) = adj(A) * A = det(A) * I

B) A * adj(A) = I

C) adj(A) * A = det(A)

D) A + adj(A) = I

24. If A is a singular matrix, what is adj(A) * A?

A) The zero matrix (O).

B) The identity matrix (I).

C) det(A) * I.

D) adj(A).

25. If A is an n x n non-singular matrix, what is the property of adj(adj(A))?

A) (det(A))^(n-2) * A

B) A

C) (det(A))^(n-1) * A

D) adj(A)

26. If det(A) = 0 for a square matrix A, then A is called a:

A) Singular matrix

B) Non-singular matrix

C) Identity matrix

D) Diagonal matrix

27. For the matrix A = [[1, 2, 3], [0, 1, 4], [5, 6, 0]], what is the determinant of A?

A) -24

B) 24

C) 1

D) 0

28. Using A = [[1, 2, 3], [0, 1, 4], [5, 6, 0]], what is the cofactor of the element a₁₁?

A) -24

B) 24

C) 1

D) 0

29. Let A = [[1, 2, 3], [0, 1, 4], [5, 6, 0]]. What is the minor of the element a₁₁?

A) -24

B) 24

C) 1

D) 0

30. If A is a 3x3 non-singular matrix, what is the order of adj(A)?

A) 3x3

B) 1x1

C) 9x9

D) 3x1

31. What is the minor of an element aᵢⱼ in a matrix A?

A) The determinant of the submatrix obtained by deleting the i-th row and j-th column of A.

B) The element aᵢⱼ itself.

C) The sum of all elements in the i-th row and j-th column.

D) The determinant of the matrix A.

32. What is the cofactor of an element aᵢⱼ in a matrix A?

A) (-1)^(i+j) * Mᵢⱼ, where Mᵢⱼ is the minor of aᵢⱼ.

B) (i+j) * Mᵢⱼ, where Mᵢⱼ is the minor of aᵢⱼ.

C) (-1)^(i*j) * Mᵢⱼ, where Mᵢⱼ is the minor of aᵢⱼ.

D) Mᵢⱼ, where Mᵢⱼ is the minor of aᵢⱼ.

33. If A is a non-singular square matrix, then (kA)⁻¹, where k is a non-zero scalar, is equal to:

A) (1/k)A⁻¹

B) kA⁻¹

C) A⁻¹/k

D) k * A⁻¹

34. If A is a non-singular square matrix, then (Aᵀ)⁻¹ is equal to:

A) (A⁻¹)ᵀ

B) A⁻¹

C) Aᵀ

D) 1 / Aᵀ

35. If A and B are non-singular square matrices of the same order, then (AB)⁻¹ is equal to:

A) B⁻¹A⁻¹

B) A⁻¹B⁻¹

C) (BA)⁻¹

D) A⁻¹ + B⁻¹

36. For A = [[2, 3], [1, 4]], what is its inverse A⁻¹?

A) [[4/5, -3/5], [-1/5, 2/5]]

B) [[5/4, -5/3], [-5/1, 5/2]]

C) [[2/5, 3/5], [1/5, 4/5]]

D) [[4, -3], [-1, 2]]

37. Using the matrix A = [[2, 3], [1, 4]], what is its adjoint?

A) [[4, -3], [-1, 2]]

B) [[2, -3], [-1, 4]]

C) [[4, 3], [1, 2]]

D) [[-2, -3], [-1, -4]]

38. Let A = [[2, 3], [1, 4]]. What is the determinant of A?

A) 5

B) 8

C) 6

D) 11

39. Let A be a 2x2 matrix [[a, b], [c, d]]. What is its adjoint?

A) [[d, -b], [-c, a]]

B) [[a, -b], [-c, d]]

C) [[d, b], [c, a]]

D) [[-a, -b], [-c, -d]]

40. If A is a singular matrix (det(A) = 0), does its inverse A⁻¹ exist?

A) No, the inverse does not exist for a singular matrix.

B) Yes, the inverse always exists.

C) The inverse exists if the matrix is also symmetric.

D) The inverse exists only if the matrix is diagonal.

41. What is the inverse of the identity matrix I of order n?

A) The identity matrix I itself.

B) The zero matrix.

C) The transpose of I.

D) n * I

42. What is the adjoint of the identity matrix I of order n?

A) The identity matrix I itself.

B) The zero matrix.

C) The transpose of I.

D) n * I

43. If A is a non-singular square matrix of order n, what is the determinant of its inverse, i.e., det(A⁻¹)?

A) 1 / det(A)

B) det(A)

C) (det(A))⁻¹

D) det(A) * n

44. What is the formula for the inverse of a non-singular square matrix A?

A) A⁻¹ = adj(A) / det(A)

B) A⁻¹ = det(A) / adj(A)

C) A⁻¹ = adj(A) * det(A)

D) A⁻¹ = I / det(A)

45. When is the inverse of a square matrix A, denoted by A⁻¹, defined?

A) When A is a non-singular matrix (det(A) ≠ 0).

B) When A is a singular matrix (det(A) = 0).

C) When A is a square matrix of any order.

D) When A is a symmetric matrix.

46. If A is a square matrix of order n, what is the determinant of its adjoint, i.e., det(adj(A))?

A) (det(A))^(n-1)

B) (det(A))^n

C) det(A)

D) 1 / det(A)

47. For a square matrix A of order n, what is the relationship between A, adj(A), and det(A)?

A) A * adj(A) = det(A) * I

B) A * adj(A) = det(A)

C) A * adj(A) = I

D) A + adj(A) = det(A) * I

48. What is the definition of the adjoint of a square matrix A?

A) The transpose of the cofactor matrix of A.

B) The matrix formed by the cofactors of the elements of A.

C) The transpose of the matrix A.

D) The matrix A multiplied by its determinant.

Adjoint and inverse of a square matrix - Question Bank