JEE Main - Adjoint and inverse of a square matrix One Line Questions & Answers

1. Let A = [[1, 2, 3], [0, 1, 4], [5, 6, 0]]. What is the minor of the element a₁₁? — -24

2. Using A = [[1, 2, 3], [0, 1, 4], [5, 6, 0]], what is the cofactor of the element a₁₁? — -24

3. For the matrix A = [[1, 2, 3], [0, 1, 4], [5, 6, 0]], what is the determinant of A? — -24

4. What is the cofactor of an element aᵢⱼ in a matrix A? — (-1)^(i+j) * Mᵢⱼ, where Mᵢⱼ is the minor of aᵢⱼ.

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

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

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

8. If A is a square matrix of order n, what is the determinant of its adjoint, i.e., det(adj(A))? — (det(A))^(n-1)

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

10. If A is an n x n non-singular matrix, what is the property of adj(adj(A))? — (det(A))^(n-2) * A

11. Let A be a 3x3 matrix. If adj(A) = [[-1, 2, 0], [0, -3, 0], [0, 0, -4]], what is A⁻¹? — [[-1/12, 1/6, 0], [0, 1/4, 0], [0, 0, -1/3]]

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

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

14. For A = [[2, 3], [1, 4]], what is its inverse A⁻¹? — [[4/5, -3/5], [-1/5, 2/5]]

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

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

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

18. Let A be a 3x3 matrix. If adj(A) = [[1, 2, 3], [4, 5, 6], [7, 8, 9]], what is det(A)? — 0

19. If A is a skew-symmetric matrix of odd order, then det(A) is: — 0

20. 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)? — 1

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

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

23. Let A be a 3x3 matrix. If det(A) = 5, what is det(adj(A))? — 25

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

25. Let A = [[2, 3], [1, 4]]. What is the determinant of A? — 5

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

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

28. Which of the following statements is always true for a non-singular matrix A? — A * adj(A) = adj(A) * A = det(A) * I

29. For a square matrix A of order n, what is the relationship between A, adj(A), and det(A)? — A * adj(A) = det(A) * I

30. If A is a skew-symmetric matrix of even order, then det(A) is: — A perfect square

31. What is the formula for the inverse of a non-singular square matrix A? — A⁻¹ = adj(A) / det(A)

32. If A is a non-singular matrix, which of the following is NOT necessarily true? — adj(A) * A⁻¹ = I

33. If A is an n x n orthogonal matrix, what is its inverse? — Aᵀ

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

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

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

37. What is the adjoint of a scalar matrix kI of order n? — k^(n-1) * I

38. If A is a singular matrix (det(A) = 0), does its inverse A⁻¹ exist? — No, the inverse does not exist for a singular matrix.

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

40. What is the minor of an element aᵢⱼ in a matrix A? — The determinant of the submatrix obtained by deleting the i-th row and j-th column of A.

41. What is the adjoint of the identity matrix I of order n? — The identity matrix I itself.

42. What is the inverse of the identity matrix I of order n? — The identity matrix I itself.

43. What is the definition of the adjoint of a square matrix A? — The transpose of the cofactor matrix of A.

44. If A is a singular matrix, what is adj(A) * A? — The zero matrix (O).

45. 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: — True

46. When is the inverse of a square matrix A, denoted by A⁻¹, defined? — When A is a non-singular matrix (det(A) ≠ 0).

47. If A is a non-singular matrix, when is adj(A) = A? — When A² = I and det(A) = 1.

48. If A is a non-singular matrix, when is adj(A) = A⁻¹? — When det(A) = 1.

Adjoint and inverse of a square matrix - One Line Questions