Practice MCQs: Matrices & Determinants

This MCQ set covers all important NCERT concepts from Matrices and Determinants, strictly based on Class 12 syllabus.

1. The order of a matrix having 3 rows and 2 columns is:
   
   (A) 2 × 3   (B) 3 × 2   (C) 3 × 3   (D) 2 × 2
2. A square matrix has:
   
   (A) equal rows only
   
   (B) equal columns only
   
   (C) equal rows and columns
   
   (D) unequal rows and columns
3. The identity matrix is a:
   
   (A) zero matrix
   
   (B) diagonal matrix
   
   (C) scalar matrix
   
   (D) both (B) and (C)
4. If A is a matrix of order 2 × 3 and B is of order 3 × 2, then AB is of order:
   
   (A) 2 × 2   (B) 3 × 3   (C) 2 × 3   (D) 3 × 2
5. The determinant of a 2 × 2 matrix
   |a b|
   |c d|
   is:
   
   (A) ad − bc   (B) ab − cd   (C) ac − bd   (D) ad + bc
6. The determinant of an identity matrix is:
   
   (A) 0   (B) 1   (C) −1   (D) Depends on order
7. If |A| = 0, then matrix A is:
   
   (A) invertible
   
   (B) singular
   
   (C) identity
   
   (D) scalar
8. The inverse of a matrix exists only if:
   
   (A) A is diagonal
   
   (B) A is square
   
   (C) |A| ≠ 0
   
   (D) both (B) and (C)
9. The determinant of a triangular matrix is equal to:
   
   (A) sum of diagonal elements
   
   (B) product of diagonal elements
   
   (C) zero
   
   (D) one
10. If A is a square matrix, then A · adj(A) equals:
    
    (A) |A|I
    
    (B) A²
    
    (C) I
    
    (D) 0
11. The determinant of a matrix changes sign if:
    
    (A) a row is multiplied by a constant
    
    (B) two rows are interchanged
    
    (C) a row is added to another row
    
    (D) determinant is transposed
12. If two rows of a matrix are identical, then its determinant is:
    
    (A) 1   (B) −1   (C) 0   (D) Undefined
13. The transpose of a matrix of order 3 × 4 is of order:
    
    (A) 3 × 4   (B) 4 × 3   (C) 3 × 3   (D) 4 × 4
14. (Aᵀ)ᵀ equals:
    
    (A) A   (B) −A   (C) I   (D) 0
15. If A is symmetric, then:
    
    (A) Aᵀ = −A
    
    (B) Aᵀ = A
    
    (C) A = I
    
    (D) |A| = 0
16. A skew-symmetric matrix satisfies:
    
    (A) Aᵀ = A
    
    (B) Aᵀ = −A
    
    (C) Aᵀ = I
    
    (D) A = 0
17. The inverse of a matrix A is denoted by:
    
    (A) A⁻¹   (B) adj A   (C) Aᵀ   (D) |A|
18. The determinant of a matrix and its transpose are:
    
    (A) equal
    
    (B) negatives
    
    (C) reciprocals
    
    (D) unrelated
19. If |A| = 5, then |2A| for a 2 × 2 matrix is:
    
    (A) 10   (B) 20   (C) 5   (D) 40
20. The matrix used to find inverse is:
    
    (A) transpose
    
    (B) identity
    
    (C) adjoint
    
    (D) zero
21. The inverse of identity matrix is:
    
    (A) zero matrix
    
    (B) identity matrix
    
    (C) diagonal matrix
    
    (D) scalar matrix
22. The determinant of a skew-symmetric matrix of odd order is:
    
    (A) 1   (B) −1   (C) 0   (D) Undefined
23. If A and B are invertible, then (AB)⁻¹ equals:
    
    (A) A⁻¹B⁻¹
    
    (B) B⁻¹A⁻¹
    
    (C) (A+B)⁻¹
    
    (D) AB
24. Which matrix has all diagonal elements equal and others zero?
    
    (A) Identity
    
    (B) Scalar
    
    (C) Zero
    
    (D) Singular
25. The determinant of a zero matrix is:
    
    (A) 1   (B) −1   (C) 0   (D) Undefined
26. The minor of an element is obtained by:
    
    (A) deleting its row and column
    
    (B) transposing matrix
    
    (C) multiplying rows
    
    (D) dividing columns
27. The cofactor of an element depends on:
    
    (A) its position
    
    (B) its value only
    
    (C) matrix order only
    
    (D) transpose
28. Which matrix has determinant always zero?
    
    (A) Identity
    
    (B) Singular
    
    (C) Scalar
    
    (D) Diagonal
29. If A is invertible, then |A⁻¹| equals:
    
    (A) |A|
    
    (B) 1/|A|
    
    (C) −|A|
    
    (D) 0
30. The determinant of a 1 × 1 matrix [a] is:
    
    (A) 0   (B) 1   (C) a   (D) −a
31. The inverse of a matrix exists only for:
    
    (A) rectangular matrices
    
    (B) square matrices
    
    (C) row matrices
    
    (D) column matrices
32. The value of |A| if A has a zero row is:
    
    (A) 1   (B) −1   (C) 0   (D) Undefined
33. Which operation does not change determinant?
    
    (A) R₁ ↔ R₂
    
    (B) R₁ → kR₁
    
    (C) R₁ → R₁ + R₂
    
    (D) None
34. The adjoint of adjoint of A equals:
    
    (A) A
    
    (B) |A|A
    
    (C) |A|A⁻¹
    
    (D) |A|ⁿ⁻²A
35. If |A| = 1, then A is:
    
    (A) singular
    
    (B) invertible
    
    (C) zero
    
    (D) symmetric
36. The determinant of a diagonal matrix is:
    
    (A) sum of diagonals
    
    (B) product of diagonals
    
    (C) zero
    
    (D) one
37. The inverse of a singular matrix is:
    
    (A) zero
    
    (B) identity
    
    (C) not defined
    
    (D) diagonal
38. Which chapter introduces inverse method of solving equations?
    
    (A) Relations
    
    (B) Determinants
    
    (C) Matrices
    
    (D) Integrals
39. For AX = B, solution exists uniquely if:
    
    (A) A is identity
    
    (B) |A| ≠ 0
    
    (C) B = 0
    
    (D) A is diagonal
40. The number of elements in a 3 × 3 matrix is:
    
    (A) 3   (B) 6   (C) 9   (D) 12
41. Transpose of a symmetric matrix is:
    
    (A) zero
    
    (B) identity
    
    (C) same matrix
    
    (D) undefined
42. Which is NOT a square matrix?
    
    (A) 2 × 2   (B) 3 × 3   (C) 1 × 1   (D) 2 × 3
43. The determinant is defined only for:
    
    (A) row matrices
    
    (B) column matrices
    
    (C) square matrices
    
    (D) diagonal matrices
44. The identity matrix is denoted by:
    
    (A) A   (B) I   (C) O   (D) E