Determinants of order two and three - Question Bank

1. If A = [[1, 2], [3, 4]] and B = [[1, 0], [0, 1]], what is det(A)?
A) -2
B) 2
C) 1
D) 5
2. What is the determinant of the matrix [[1, 0, 0], [2, 3, 0], [4, 5, 6]]?
A) 0
B) 1
C) 6
D) 18
3. If det(A) = 3 for a 3x3 matrix A, what is det(A^T)?
A) 1/3
B) -3
C) 3
D) 9
4. What is the determinant of the matrix [[1, 2], [2, 4]]?
A) 0
B) 1
C) 4
D) 5
5. 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
6. What is the determinant of the matrix [[1, 0, 0], [0, 1, 0], [0, 0, 1]]?
A) 0
B) 1
C) 3
D) -1
7. If A is a 2x2 matrix with det(A) = 7, what is det(2A)?
A) 14
B) 28
C) 4
D) 49
8. 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
9. If A = [[2, 0], [0, 3]], what is det(A^3)?
A) 6
B) 18
C) 36
D) 216
10. 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
11. 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
12. What is the determinant of the matrix [[1, 0, 5], [0, 1, 6], [0, 0, 1]]?
A) 0
B) 1
C) 5
D) 6
13. 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
14. What is the determinant of the matrix [[1, 2, 3], [1, 2, 3], [4, 5, 6]]?
A) 0
B) 1
C) 6
D) 12
15. If A = [[1, 2], [3, 4]] and B = [[0, 1], [1, 0]], what is det(AB)?
A) -2
B) 2
C) 0
D) -4
16. 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
17. 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
18. If A = [[1, 2, 3], [0, 0, 0], [4, 5, 6]], what is det(A)?
A) 1
B) 0
C) 3
D) 12
19. 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)
20. If A is a 3x3 matrix and det(A) = 10, what is det(2A)?
A) 20
B) 40
C) 80
D) 100
21. What is the determinant of the matrix [[0, 0, 0], [1, 2, 3], [4, 5, 6]]?
A) 1
B) -1
C) 0
D) 12
22. 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
23. 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
24. If A is a 2x2 matrix with det(A) = -3, what is det(-A)?
A) 3
B) -3
C) 9
D) -9
25. What is the determinant of the matrix [[1, 1, 1], [1, 2, 3], [1, 4, 9]]?
A) 0
B) 1
C) 2
D) 6
26. If A = [[2, 1], [1, 2]], what is det(A^2)?
A) 3
B) 9
C) 81
D) 6
27. 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
28. What is the determinant of the matrix [[1, 2], [3, 4]]?
A) 10
B) -2
C) 2
D) 5
29. 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
30. 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
31. 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)
32. 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
33. 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
34. If det(A) = 5 for a 2x2 matrix A, what is det(3A)?
A) 15
B) 45
C) 9
D) 125
35. What is the determinant of the matrix [[5, 0], [0, 5]]?
A) 0
B) 5
C) 10
D) 25
36. 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
37. 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
38. 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
39. 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)
40. 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
41. 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)
42. 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
43. 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
44. 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
45. 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)
46. 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)
47. 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
48. What is the determinant of the identity matrix of order 2, I = [[1, 0], [0, 1]]?
A) 0
B) 1
C) 2
D) -1
49. If A = [[2, 3], [4, 5]], what is the determinant of A, denoted as |A|?
A) 2
B) -2
C) 10
D) 23
50. 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
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