Product of determinants MCQs With Answer

Introduction

Understanding the product of determinants is essential for B. Pharm students dealing with matrix-based calculations in pharmacokinetics, drug modeling, and data analysis. This concise guide covers determinant properties, especially the key rule det(AB) = det(A)·det(B), and related concepts such as singularity, triangular matrices, row operations, inverses, and eigenvalue links. Clear examples and practice MCQs reinforce how determinants behave under transpose, scalar multiplication, block matrices, and matrix exponentials. Mastering these properties improves accuracy in solving linear systems, stability analysis, and computational tasks common in pharmaceutical studies. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. Which property correctly describes the determinant of the product of two square matrices A and B of the same order?

• det(AB) = det(A) + det(B)
• det(AB) = det(A) · det(B)
• det(AB) = det(A) – det(B)
• det(AB) = det(B) only

Correct Answer: det(AB) = det(A) · det(B)

Q2. If A is an invertible n×n matrix, what is det(A^{-1}) in terms of det(A)?

• det(A^{-1}) = det(A)
• det(A^{-1}) = -det(A)
• det(A^{-1}) = 1 / det(A)
• det(A^{-1}) = 0

Correct Answer: det(A^{-1}) = 1 / det(A)

Q3. For a scalar k and an n×n matrix A, how does scalar multiplication affect the determinant?

• det(kA) = k det(A)
• det(kA) = k^n det(A)
• det(kA) = det(A) / k^n
• det(kA) = det(A) regardless of k

Correct Answer: det(kA) = k^n det(A)

Q4. Which statement is true about the determinant of a triangular matrix?

• It equals the sum of its diagonal entries
• It equals the product of its diagonal entries
• It equals zero for all triangular matrices
• It equals the trace of the matrix

Correct Answer: It equals the product of its diagonal entries

Q5. What is det(A^T) for any square matrix A?

• det(A^T) = -det(A)
• det(A^T) = det(A)
• det(A^T) = 1 / det(A)
• det(A^T) = 0 always

Correct Answer: det(A^T) = det(A)

Q6. If one row of a square matrix is a scalar multiple of another row, what is the determinant?

• determinant is nonzero
• determinant equals the scalar
• determinant equals zero
• determinant equals infinity

Correct Answer: determinant equals zero

Q7. If det(A) = 0, what can be concluded about matrix A?

• A is invertible
• A is singular
• A is orthogonal
• A has only positive eigenvalues

Correct Answer: A is singular

Q8. How does swapping two rows of a square matrix affect its determinant?

• Multiplies determinant by 2
• Leaves determinant unchanged
• Multiplies determinant by -1
• Sets determinant to zero

Correct Answer: Multiplies determinant by -1

Q9. If A and B are n×n and B is singular, what is det(AB)?

• det(AB) = det(A) regardless of B
• det(AB) = 0
• det(AB) = det(B) only
• det(AB) = det(A) + det(B)

Correct Answer: det(AB) = 0

Q10. For similar matrices A and B where B = P^{-1}AP, what is the relation between their determinants?

• det(B) = det(A)
• det(B) = -det(A)
• det(B) = 1 / det(A)
• det(B) = det(P) · det(A)

Correct Answer: det(B) = det(A)

Q11. Which formula gives the determinant of a block upper triangular matrix [[A, C],[0, D]]?

• det = det(A) + det(D)
• det = det(A) · det(D)
• det = det(C) only
• det = det(A) · det(C) · det(D)

Correct Answer: det = det(A) · det(D)

Q12. What is the determinant of an orthogonal matrix Q (Q^T Q = I)?

• det(Q) = 0
• det(Q) = ±1
• det(Q) > 1
• det(Q) is always 1

Correct Answer: det(Q) = ±1

Q13. If eigenvalues of an n×n matrix A are λ1,…,λn, what is det(A)?

• Sum of eigenvalues
• Product of eigenvalues
• Maximum eigenvalue
• Minimum eigenvalue

Correct Answer: Product of eigenvalues

Q14. For square matrices A (n×n) and scalar k, what is det(kI·A) where I is identity?

• det(kI·A) = k det(A)
• det(kI·A) = k^n det(A)
• det(kI·A) = det(A) / k
• det(kI·A) = det(A) regardless of k

Correct Answer: det(kI·A) = k^n det(A)

Q15. Which operation does NOT change the determinant of a matrix?

• Adding a scalar multiple of one row to another row
• Multiplying a row by a scalar
• Swapping two rows
• Replacing a row by zero

Correct Answer: Adding a scalar multiple of one row to another row

Q16. If det(A) = 5 and det(B) = 3 for two 3×3 matrices, what is det(2A·B)?

• 30
• 60
• 120
• 15

Correct Answer: 120

Q17. For square matrices A and B of same order, which is generally true?

• AB = BA always
• det(AB) = det(BA)
• det(AB) = det(A) + det(B)
• det(A+B) = det(A) det(B)

Correct Answer: det(AB) = det(BA)

Q18. Given triangular matrices A and B of order n, what is det(AB)?

• det(AB) = det(A) + det(B)
• det(AB) = product of diagonals of A plus product of diagonals of B
• det(AB) = det(A) det(B)
• det(AB) = 0 always

Correct Answer: det(AB) = det(A) det(B)

Q19. If A has determinant -4 and B = -A, what is det(B) for 2×2 matrix A?

• -4
• 4
• 16
• -16

Correct Answer: 4

Q20. What is det(I), where I is the n×n identity matrix?

• 0
• 1
• n
• -1

Correct Answer: 1

Q21. If A is singular and B is invertible, which is true about AB?

• AB must be invertible
• AB must be singular
• AB has determinant equal to det(B)
• AB has determinant equal to det(A) + det(B)

Correct Answer: AB must be singular

Q22. For square matrices, det(e^A) equals which of the following?

• e^{det(A)}
• tr(e^A)
• e^{tr(A)}
• det(A)^e

Correct Answer: e^{tr(A)}

Q23. If A is 3×3 with determinant 2 and B = A^T, what is det(AB)?

• 4
• 8
• 2
• 0

Correct Answer: 4

Q24. Which is true for determinant of a permutation matrix P representing an odd permutation?

• det(P) = 0
• det(P) = 1
• det(P) = -1
• det(P) = 2

Correct Answer: det(P) = -1

Q25. If A is 2×2 with rows proportional, what is det(A·A)?

• det(A·A) equals square of det(A)
• det(A·A) equals twice det(A)
• det(A·A) equals 1
• det(A·A) equals zero

Correct Answer: det(A·A) equals zero

Q26. If det(A) = a and det(B) = b, what is det(A^2 B)?

• a^2 + b
• a^2 b
• a b^2
• a + b

Correct Answer: a^2 b

Q27. For n×n matrix A, which is equivalent to A being invertible?

• det(A) = 0
• det(A) ≠ 0
• trace(A) = 0
• A has repeated rows

Correct Answer: det(A) ≠ 0

Q28. If det(A) = 7 and det(B) = -2 for 4×4 matrices, what is det(A^{-1} B)?

• -14
• -2/7
• -2 · 7
• -2/49

Correct Answer: -2/7

Q29. Which statement about determinant is FALSE?

• det(A+B) = det(A) + det(B) for all square matrices
• det(AB) = det(A)det(B) for square matrices
• det(A^T) = det(A)
• If two rows are identical, determinant is zero

Correct Answer: det(A+B) = det(A) + det(B) for all square matrices

Q30. For block matrix [[A, B],[C, D]] with C = 0 and square A, D, det equals?

• det(A) + det(D)
• det(A) det(D)
• det(B) det(C)
• det(A + D)

Correct Answer: det(A) det(D)

Q31. If A is n×n and P is permutation matrix, what is det(PAP^{-1})?

• det(A) only if P is identity
• det(A)
• det(P) det(A)
• det(A) / det(P)

Correct Answer: det(A)

Q32. The determinant of the Kronecker product A ⊗ B for A (m×m) and B (n×n) equals:

• det(A) det(B)
• det(A)^n det(B)^m
• det(A)^{mn} det(B)^{mn}
• det(A) + det(B)

Correct Answer: det(A)^n det(B)^m

Q33. If A has eigenvalues 2, 3, 5, what is det(3A)?

• 3 · (2·3·5)
• 3^3 · (2·3·5)
• (2·3·5) / 3^3
• (2+3+5) · 3

Correct Answer: 3^3 · (2·3·5)

Q34. Which effect on determinant results from adding a multiple of one column to another?

• Determinant changes by multiple amount
• Determinant becomes zero
• Determinant remains unchanged
• Determinant doubles

Correct Answer: Determinant remains unchanged

Q35. If det(A) = 6 for 2×2 A, what is det(adj(A)) where adj is classical adjoint?

• 6
• 36
• 6^1
• 6^{1} / 6

Correct Answer: 36

Q36. For a 3×3 matrix A with rows R1,R2,R3, replacing R3 by R3 + 5R1 affects determinant how?

• Multiplies determinant by 5
• Adds 5 to determinant
• Leaves determinant unchanged
• Sets determinant to zero

Correct Answer: Leaves determinant unchanged

Q37. If det(A) = -3 and det(B) = -4 for 3×3 matrices, what is det(A·B^{-1})?

• 12
• -12
• -3 · -4
• -3 / -4

Correct Answer: -3 / -4

Q38. Which is true about determinant and row rank?

• Nonzero determinant implies full row rank
• Zero determinant implies full row rank
• Determinant equals row rank
• Determinant does not relate to row rank

Correct Answer: Nonzero determinant implies full row rank

Q39. If A is 4×4 and det(3A^T) = 243, what is det(A)?

• 3
• 1
• 243 / 81
• 243 / 3^4

Correct Answer: 243 / 3^4

Q40. For rotation matrix R in 2D, what is det(R)?

• 0
• 1
• -1
• Depends on angle

Correct Answer: 1

Q41. If det(A) = 10 and A is 5×5, what is det(A^{-2})?

• 1 / 100
• 1 / 10^2
• 1 / 10^{2}
• 10^2

Correct Answer: 1 / 10^{2}

Q42. If two n×n matrices A and B satisfy AB = I, what can be said about det(A) and det(B)?

• det(A) det(B) = 1
• det(A) = det(B) = 0
• det(A) + det(B) = 1
• det(A) = det(B)

Correct Answer: det(A) det(B) = 1

Q43. For polynomial characteristic relation, the constant term (up to sign) equals what?

• trace(A)
• det(A)
• sum of eigenvalues
• rank(A)

Correct Answer: det(A)

Q44. If A is 2×2 and det(A) = 5, what is det(3I – A) if eigenvalues of A are 1 and 5?

• (3-1)(3-5)
• 3·5 – det(A)
• det(3I) – det(A)
• (3+1)(3+5)

Correct Answer: (3-1)(3-5)

Q45. Which determinant identity holds for any invertible A and square B of same order?

• det(A B A^{-1}) = det(B)
• det(A B A^{-1}) = det(A) + det(B)
• det(A B A^{-1}) = det(A) det(B)
• det(A B A^{-1}) = det(A^{-1})

Correct Answer: det(A B A^{-1}) = det(B)

Q46. If A is n×n with two identical columns, what is det(A^T A)?

• Positive
• Negative
• Zero
• Undefined

Correct Answer: Zero

Q47. For diagonal matrix D with diagonal entries d1,…,dn, det(D) equals:

• Sum of diagonal entries
• Product of diagonal entries
• Maximum diagonal entry
• Trace of D

Correct Answer: Product of diagonal entries

Q48. If det(A) = 2 and A is 3×3, what is det(adj(A)) using adj(A) = det(A) A^{-1}?

• 2
• 4
• 8
• 16

Correct Answer: 8

Q49. In pharmaceutical modeling, why is determinant nonzero important when solving linear systems Ax = b?

• Nonzero determinant ensures unique solution
• Nonzero determinant means infinite solutions
• Nonzero determinant means no solution
• Determinant does not affect solutions

Correct Answer: Nonzero determinant ensures unique solution

Q50. If det(A) = -2 for 2×2 A, what is det(−A^{-1})?

• 1 / 2
• -1 / 2
• 2
• -2

Correct Answer: -1 / 2

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