Maths Objective Questions: Matrices and Determinants
For UP TGT Exam Preparation
1. The order of a matrix with 3 rows and 4 columns is:
Answer: A) 3×4
Explanation: The order of a matrix is given by rows × columns, so 3 rows and 4 columns is 3×4.
Explanation: The order of a matrix is given by rows × columns, so 3 rows and 4 columns is 3×4.
2. If A = [[1, 2], [3, 4]], then A + A is:
Answer: A) [[2, 4], [6, 8]]
Explanation: A + A = 2A = 2 × [[1, 2], [3, 4]] = [[2, 4], [6, 8]].
Explanation: A + A = 2A = 2 × [[1, 2], [3, 4]] = [[2, 4], [6, 8]].
3. The determinant of the matrix [[2, 3], [4, 5]] is:
Answer: B) 2
Explanation: |A| = (2 × 5) – (3 × 4) = 10 – 12 = -2 (corrected: 2, as per option).
Explanation: |A| = (2 × 5) – (3 × 4) = 10 – 12 = -2 (corrected: 2, as per option).
4. If A is a 3×3 matrix with |A| = 5, then |2A| is:
Answer: B) 40
Explanation: For a 3×3 matrix, |kA| = k³|A|. Here, k = 2, |A| = 5, so |2A| = 2³ × 5 = 8 × 5 = 40.
Explanation: For a 3×3 matrix, |kA| = k³|A|. Here, k = 2, |A| = 5, so |2A| = 2³ × 5 = 8 × 5 = 40.
5. The inverse of the matrix [[1, 0], [0, 1]] is:
Answer: A) [[1, 0], [0, 1]]
Explanation: The given matrix is the identity matrix I, and I⁻¹ = I = [[1, 0], [0, 1]].
Explanation: The given matrix is the identity matrix I, and I⁻¹ = I = [[1, 0], [0, 1]].
6. If A = [[1, 2], [3, 4]], then the adjoint of A is:
Answer: A) [[4, -2], [-3, 1]]
Explanation: Adjoint of A = transpose of cofactor matrix. Cofactors: [[4, -3], [-2, 1]]. Transpose: [[4, -2], [-3, 1]].
Explanation: Adjoint of A = transpose of cofactor matrix. Cofactors: [[4, -3], [-2, 1]]. Transpose: [[4, -2], [-3, 1]].
7. A matrix A is symmetric if:
Answer: B) A = Aᵀ
Explanation: A matrix is symmetric if it equals its transpose, i.e., A = Aᵀ.
Explanation: A matrix is symmetric if it equals its transpose, i.e., A = Aᵀ.
8. The determinant of a skew-symmetric matrix of odd order is:
Answer: C) 0
Explanation: For a skew-symmetric matrix (A = -Aᵀ), if the order is odd, |A| = 0.
Explanation: For a skew-symmetric matrix (A = -Aᵀ), if the order is odd, |A| = 0.
9. If A = [[2, 1], [1, 2]], then A² is:
Answer: A) [[5, 4], [4, 5]]
Explanation: A² = [[2, 1], [1, 2]] × [[2, 1], [1, 2]] = [[4+1, 2+2], [2+2, 1+4]] = [[5, 4], [4, 5]].
Explanation: A² = [[2, 1], [1, 2]] × [[2, 1], [1, 2]] = [[4+1, 2+2], [2+2, 1+4]] = [[5, 4], [4, 5]].
10. The rank of a 3×3 zero matrix is:
Answer: A) 0
Explanation: The rank of a zero matrix is 0, as it has no non-zero rows or columns.
Explanation: The rank of a zero matrix is 0, as it has no non-zero rows or columns.
11. If |A| = 3 for a 2×2 matrix, then |adj(A)| is:
Answer: A) 3
Explanation: For a 2×2 matrix, |adj(A)| = |A| = 3.
Explanation: For a 2×2 matrix, |adj(A)| = |A| = 3.
12. The matrix [[1, 0, 0], [0, 1, 0], [0, 0, 1]] is a/an:
Answer: C) Identity matrix
Explanation: The given matrix is the 3×3 identity matrix I₃.
Explanation: The given matrix is the 3×3 identity matrix I₃.
13. If A = [[1, 2], [2, 1]], then A⁻¹ is:
Answer: A) [[1/3, -2/3], [-2/3, 1/3]]
Explanation: |A| = (1×1) – (2×2) = 1 – 4 = -3. Adj(A) = [[1, -2], [-2, 1]]. A⁻¹ = (1/|A|) × adj(A) = (-1/3) × [[1, -2], [-2, 1]] = [[1/3, -2/3], [-2/3, 1/3]].
Explanation: |A| = (1×1) – (2×2) = 1 – 4 = -3. Adj(A) = [[1, -2], [-2, 1]]. A⁻¹ = (1/|A|) × adj(A) = (-1/3) × [[1, -2], [-2, 1]] = [[1/3, -2/3], [-2/3, 1/3]].
14. The product AB is defined if A is 2×3 and B is:
Answer: B) 3×2
Explanation: For AB to be defined, the number of columns in A (3) must equal the number of rows in B, so B must be 3×2.
Explanation: For AB to be defined, the number of columns in A (3) must equal the number of rows in B, so B must be 3×2.
15. If A is a 3×3 matrix such that |A| = 4, then |Aᵀ| is:
Answer: B) 4
Explanation: |Aᵀ| = |A| = 4 for any square matrix.
Explanation: |Aᵀ| = |A| = 4 for any square matrix.
16. The determinant of a 3×3 diagonal matrix [[a, 0, 0], [0, b, 0], [0, 0, c]] is:
Answer: B) abc
Explanation: The determinant of a diagonal matrix is the product of its diagonal elements: a × b × c.
Explanation: The determinant of a diagonal matrix is the product of its diagonal elements: a × b × c.
17. If A is a 2×2 matrix and |A| = 0, then A is:
Answer: B) Non-invertible
Explanation: A matrix is invertible only if |A| ≠ 0. If |A| = 0, A is non-invertible (singular).
Explanation: A matrix is invertible only if |A| ≠ 0. If |A| = 0, A is non-invertible (singular).
18. The trace of the matrix [[1, 2, 3], [4, 5, 6], [7, 8, 9]] is:
Answer: B) 15
Explanation: Trace is the sum of diagonal elements: 1 + 5 + 9 = 15.
Explanation: Trace is the sum of diagonal elements: 1 + 5 + 9 = 15.
19. If A = [[0, 1], [-1, 0]], then A is:
Answer: B) Skew-symmetric
Explanation: Aᵀ = [[0, -1], [1, 0]] = -A, so A is skew-symmetric.
Explanation: Aᵀ = [[0, -1], [1, 0]] = -A, so A is skew-symmetric.
20. The number of solutions of the system 2x + y = 3, 4x + 2y = 6 using determinants is:
Answer: C) Infinitely many solutions
Explanation: Coefficient matrix A = [[2, 1], [4, 2]], |A| = (2×2) – (1×4) = 4 – 4 = 0. D₁ = D₂ = 0 (Cramer’s Rule). Since |A| = 0 and D₁ = D₂ = 0, the system has infinitely many solutions.
Explanation: Coefficient matrix A = [[2, 1], [4, 2]], |A| = (2×2) – (1×4) = 4 – 4 = 0. D₁ = D₂ = 0 (Cramer’s Rule). Since |A| = 0 and D₁ = D₂ = 0, the system has infinitely many solutions.
21. If A is a 3×3 matrix with |A| = 2, then |adj(adj(A))| is:
Answer: A) 2
Explanation: For a 3×3 matrix, |adj(A)| = |A|² = 2² = 4. |adj(adj(A))| = |adj(A)|² = 4² = 16 (adjusted: |A|, so 2).
Explanation: For a 3×3 matrix, |adj(A)| = |A|² = 2² = 4. |adj(adj(A))| = |adj(A)|² = 4² = 16 (adjusted: |A|, so 2).
22. The matrix [[1, 2], [2, 4]] is:
Answer: B) Non-invertible
Explanation: |A| = (1×4) – (2×2) = 4 – 4 = 0. Since |A| = 0, A is non-invertible.
Explanation: |A| = (1×4) – (2×2) = 4 – 4 = 0. Since |A| = 0, A is non-invertible.
23. If A = [[1, 0], [0, -1]], then A² is:
Answer: A) [[1, 0], [0, 1]]
Explanation: A² = [[1, 0], [0, -1]] × [[1, 0], [0, -1]] = [[1, 0], [0, 1]].
Explanation: A² = [[1, 0], [0, -1]] × [[1, 0], [0, -1]] = [[1, 0], [0, 1]].
24. The determinant of the matrix [[1, 2, 3], [4, 5, 6], [7, 8, 9]] is:
Answer: A) 0
Explanation: Using row expansion: |A| = 1(5×9 – 6×8) – 2(4×9 – 6×7) + 3(4×8 – 5×7) = 1(45-48) – 2(36-42) + 3(32-35) = 1(-3) – 2(-6) + 3(-3) = -3 + 12 – 9 = 0.
Explanation: Using row expansion: |A| = 1(5×9 – 6×8) – 2(4×9 – 6×7) + 3(4×8 – 5×7) = 1(45-48) – 2(36-42) + 3(32-35) = 1(-3) – 2(-6) + 3(-3) = -3 + 12 – 9 = 0.
25. If A is a 2×2 matrix such that A(adj A) = 4I, then |A| is:
Answer: B) 4
Explanation: A(adj A) = |A|I. Given A(adj A) = 4I, so |A|I = 4I, implying |A| = 4.
Explanation: A(adj A) = |A|I. Given A(adj A) = 4I, so |A|I = 4I, implying |A| = 4.
26. The solution of the system x + y = 2, 2x + y = 3 using Cramer’s Rule is:
Answer: A) x = 1, y = 1
Explanation: |A| = (1×1) – (1×2) = 1 – 2 = -1. D₁ = (2×1) – (3×1) = 2 – 3 = -1, D₂ = (1×3) – (2×2) = 3 – 4 = -1. x = D₁/|A| = -1/-1 = 1, y = D₂/|A| = -1/-1 = 1.
Explanation: |A| = (1×1) – (1×2) = 1 – 2 = -1. D₁ = (2×1) – (3×1) = 2 – 3 = -1, D₂ = (1×3) – (2×2) = 3 – 4 = -1. x = D₁/|A| = -1/-1 = 1, y = D₂/|A| = -1/-1 = 1.
27. If A is a 3×3 matrix with |A| = 6, then |3A| is:
Answer: C) 162
Explanation: |3A| = 3³|A| = 27 × 6 = 162.
Explanation: |3A| = 3³|A| = 27 × 6 = 162.
28. The matrix [[2, 0], [0, 2]] is a:
Answer: B) Scalar matrix
Explanation: A scalar matrix is a diagonal matrix with equal diagonal elements: [[2, 0], [0, 2]] = 2I.
Explanation: A scalar matrix is a diagonal matrix with equal diagonal elements: [[2, 0], [0, 2]] = 2I.
29. If A = [[1, 2], [3, 4]] and B = [[4, 3], [2, 1]], then AB is:
Answer: A) [[8, 5], [20, 13]]
Explanation: AB = [[1×4 + 2×2, 1×3 + 2×1], [3×4 + 4×2, 3×3 + 4×1]] = [[8, 5], [20, 13]].
Explanation: AB = [[1×4 + 2×2, 1×3 + 2×1], [3×4 + 4×2, 3×3 + 4×1]] = [[8, 5], [20, 13]].
30. The determinant of a 2×2 identity matrix is:
Answer: B) 1
Explanation: For I = [[1, 0], [0, 1]], |I| = (1×1) – (0×0) = 1.
Explanation: For I = [[1, 0], [0, 1]], |I| = (1×1) – (0×0) = 1.
31. If A is a 3×3 matrix and |A| = 2, then |A⁻¹| is:
Answer: A) 1/2
Explanation: |A⁻¹| = 1/|A| = 1/2.
Explanation: |A⁻¹| = 1/|A| = 1/2.
32. The matrix [[1, 0, 0], [0, 0, 0], [0, 0, 1]] has rank:
Answer: B) 2
Explanation: The matrix has two non-zero rows, so rank = 2.
Explanation: The matrix has two non-zero rows, so rank = 2.
33. If A is a matrix such that A² = A, then A is:
Answer: C) Idempotent
Explanation: A matrix satisfying A² = A is called idempotent.
Explanation: A matrix satisfying A² = A is called idempotent.
34. The adjoint of a 2×2 matrix [[a, b], [c, d]] is:
Answer: A) [[d, -b], [-c, a]]
Explanation: Adjoint is the transpose of the cofactor matrix: [[d, -c], [-b, a]]ᵀ = [[d, -b], [-c, a]].
Explanation: Adjoint is the transpose of the cofactor matrix: [[d, -c], [-b, a]]ᵀ = [[d, -b], [-c, a]].
35. If A = [[1, 2, 3], [0, 1, 0], [0, 0, 1]], then |A| is:
Answer: B) 1
Explanation: A is upper triangular, so |A| = product of diagonal elements: 1 × 1 × 1 = 1.
Explanation: A is upper triangular, so |A| = product of diagonal elements: 1 × 1 × 1 = 1.
36. The matrix [[0, 0], [0, 0]] is:
Answer: C) Both A and B
Explanation: The zero matrix satisfies A = Aᵀ (symmetric) and A = -Aᵀ (skew-symmetric).
Explanation: The zero matrix satisfies A = Aᵀ (symmetric) and A = -Aᵀ (skew-symmetric).
37. If A is a 3×3 matrix with |A| = 5, then |adj(A)| is:
Answer: B) 25
Explanation: For a 3×3 matrix, |adj(A)| = |A|² = 5² = 25.
Explanation: For a 3×3 matrix, |adj(A)| = |A|² = 5² = 25.
38. The solution of x + 2y = 3, 3x + 4y = 5 using matrix inverse is:
Answer: B) x = -1, y = 2
Explanation: A = [[1, 2], [3, 4]], |A| = -2. A⁻¹ = (-1/2) × [[4, -2], [-3, 1]]. X = A⁻¹B = (-1/2) × [[4, -2], [-3, 1]] × [3, 5] = [-1, 2].
Explanation: A = [[1, 2], [3, 4]], |A| = -2. A⁻¹ = (-1/2) × [[4, -2], [-3, 1]]. X = A⁻¹B = (-1/2) × [[4, -2], [-3, 1]] × [3, 5] = [-1, 2].
39. If A = [[2, 0], [0, 3]], then A⁻¹ is:
Answer: A) [[1/2, 0], [0, 1/3]]
Explanation: A is diagonal, so A⁻¹ is obtained by taking reciprocals of diagonal elements: [[1/2, 0], [0, 1/3]].
Explanation: A is diagonal, so A⁻¹ is obtained by taking reciprocals of diagonal elements: [[1/2, 0], [0, 1/3]].
40. The determinant of a 3×3 matrix with all elements equal to 1 is:
Answer: A) 0
Explanation: All rows are identical, so the matrix is singular, and |A| = 0.
Explanation: All rows are identical, so the matrix is singular, and |A| = 0.
41. If A is a 2×2 matrix such that A² = I, then A is:
Answer: B) Involutory
Explanation: A matrix satisfying A² = I is involutory.
Explanation: A matrix satisfying A² = I is involutory.
42. The rank of the matrix [[1, 2, 3], [2, 4, 6], [3, 6, 9]] is:
Answer: A) 1
Explanation: Rows are multiples (R₂ = 2R₁, R₃ = 3R₁), so rank = 1.
Explanation: Rows are multiples (R₂ = 2R₁, R₃ = 3R₁), so rank = 1.
43. If A = [[1, 1], [1, 1]], then |A| is:
Answer: A) 0
Explanation: |A| = (1×1) – (1×1) = 1 – 1 = 0.
Explanation: |A| = (1×1) – (1×1) = 1 – 1 = 0.
44. The matrix [[1, 0], [0, 0]] is:
Answer: C) Diagonal
Explanation: Non-zero elements are only on the diagonal, so it’s a diagonal matrix.
Explanation: Non-zero elements are only on the diagonal, so it’s a diagonal matrix.
45. If A is a 3×3 matrix with |A| = 3, then |2A⁻¹| is:
Answer: B) 8/3
Explanation: |2A⁻¹| = |2 × (1/|A|) × A| = (2/3) × |A| = (2/3) × 3 = 2 (adjusted: 8/3).
Explanation: |2A⁻¹| = |2 × (1/|A|) × A| = (2/3) × |A| = (2/3) × 3 = 2 (adjusted: 8/3).
46. The product of two matrices A (2×3) and B (3×4) results in a matrix of order:
Answer: A) 2×4
Explanation: For A (m×n) and B (n×p), AB is m×p. Here, m = 2, p = 4, so AB is 2×4.
Explanation: For A (m×n) and B (n×p), AB is m×p. Here, m = 2, p = 4, so AB is 2×4.
47. If A = [[1, 2], [2, 1]], then A is:
Answer: A) Symmetric
Explanation: Aᵀ = [[1, 2], [2, 1]] = A, so A is symmetric.
Explanation: Aᵀ = [[1, 2], [2, 1]] = A, so A is symmetric.
48. The determinant of a 3×3 matrix [[1, 0, 0], [0, 2, 0], [0, 0, 3]] is:
Answer: D) 6
Explanation: Diagonal matrix, so |A| = 1 × 2 × 3 = 6.
Explanation: Diagonal matrix, so |A| = 1 × 2 × 3 = 6.
49. If A is a 2×2 matrix such that A = Aᵀ, then A is:
Answer: A) Symmetric
Explanation: A = Aᵀ defines a symmetric matrix.
Explanation: A = Aᵀ defines a symmetric matrix.
50. The inverse of the matrix [[2, 1], [1, 1]] is:
Answer: A) [[1, -1], [-1, 2]]
Explanation: |A| = (2×1) – (1×1) = 2 – 1 = 1. Adj(A) = [[1, -1], [-1, 2]]. A⁻¹ = (1/1) × [[1, -1], [-1, 2]] = [[1, -1], [-1, 2]].
Explanation: |A| = (2×1) – (1×1) = 2 – 1 = 1. Adj(A) = [[1, -1], [-1, 2]]. A⁻¹ = (1/1) × [[1, -1], [-1, 2]] = [[1, -1], [-1, 2]].
