Properties of determinants MCQs With Answer

Introduction: The Properties of determinants MCQs With Answer provide B. Pharm students a focused review of matrix determinant concepts essential for pharmacokinetics, compartmental modeling, and solving linear systems in drug formulation studies. This concise, keyword-rich guide covers determinant definitions, row and column operations, cofactor expansion, multiplicativity, triangular and diagonal matrix rules, and applications like Cramer’s rule and Jacobian transforms. Designed to reinforce analytical skills and exam readiness, the questions emphasize practical computation and theoretical properties relevant to pharmaceutical problems. Clear explanations and answers help build confidence in using matrix determinants for stability analysis and system solutions. ‘Now let’s test your knowledge with 50 MCQs on this topic.’

Q1. What is the determinant of a 2×2 matrix [[a,b],[c,d]]?

• ad + bc
• ad – bc
• ab – cd
• ac – bd

Correct Answer: ad – bc

Q2. For an n x n upper triangular matrix, which statement about its determinant is true?

• It equals the sum of diagonal entries
• It equals the product of diagonal entries
• It equals the trace raised to n
• It is always zero

Correct Answer: It equals the product of diagonal entries

Q3. If two rows of a square matrix are interchanged, how does the determinant change?

• It doubles
• It becomes zero
• It changes sign (multiplied by -1)
• It remains unchanged

Correct Answer: It changes sign (multiplied by -1)

Q4. Adding a multiple of one row to another row results in what change to the determinant?

• The determinant is multiplied by the scalar
• The determinant becomes zero
• The determinant remains unchanged
• The determinant changes sign

Correct Answer: The determinant remains unchanged

Q5. Multiplying a single row of an n x n matrix by scalar k results in what effect on the determinant?

• Determinant multiplied by k
• Determinant multiplied by k^n
• Determinant multiplied by 1/k
• Determinant unchanged

Correct Answer: Determinant multiplied by k

Q6. For an n x n matrix A and scalar k, what is det(kA)?

• k det(A)
• k^n det(A)
• det(A)^k
• det(A) / k^n

Correct Answer: k^n det(A)

Q7. Which property correctly describes det(AB) for two n x n matrices A and B?

• det(AB) = det(A) + det(B)
• det(AB) = det(A) det(B)
• det(AB) = det(A – B)
• det(AB) = det(A) / det(B)

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

Q8. What is det(A^T) relative to det(A) 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

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

Q9. What is the determinant of the identity matrix I_n?

• 0
• n
• 1
• -1

Correct Answer: 1

Q10. If det(A) = 0 for a square matrix A, which conclusion is correct?

• A is invertible
• A is singular (not invertible)
• A has full rank
• A has a unique solution for all Ax = b

Correct Answer: A is singular (not invertible)

Q11. If A is invertible, what is det(A^{-1})?

• det(A)
• -det(A)
• 1 / det(A)
• 0

Correct Answer: 1 / det(A)

Q12. Which expression defines the cofactor C_{ij} of element a_{ij}?

• C_{ij} = a_{ij} * det(A)
• C_{ij} = (-1)^{i+j} * M_{ij}, where M_{ij} is the minor
• C_{ij} = det(A) / a_{ij}
• C_{ij} = sum of row i plus column j

Correct Answer: C_{ij} = (-1)^{i+j} * M_{ij}, where M_{ij} is the minor

Q13. What is the Laplace expansion (cofactor expansion) used for?

• Finding eigenvalues only
• Computing determinant by expanding along a row or column
• Solving nonlinear equations
• Diagonalizing matrices

Correct Answer: Computing determinant by expanding along a row or column

Q14. How does the determinant change under multiplication by a permutation matrix P that permutes rows?

• It remains unchanged
• It is multiplied by the sign of the permutation
• It becomes its reciprocal
• It becomes zero

Correct Answer: It is multiplied by the sign of the permutation

Q15. Which statement connects determinant and eigenvalues λ_i of an n x n matrix A?

• det(A) equals the sum of eigenvalues
• det(A) equals the product of eigenvalues (with multiplicity)
• det(A) equals the largest eigenvalue
• det(A) equals the trace squared

Correct Answer: det(A) equals the product of eigenvalues (with multiplicity)

Q16. For a block upper triangular matrix with square diagonal blocks, what is the determinant?

• Sum of determinants of diagonal blocks
• Product of determinants of diagonal blocks
• Determinant of top-left block only
• Always zero

Correct Answer: Product of determinants of diagonal blocks

Q17. Which property implies that if two rows of a matrix are identical, the determinant is zero?

• Multilinearity
• Alternating property of the determinant
• Homogeneity of degree n
• Multiplicativity

Correct Answer: Alternating property of the determinant

Q18. What is the determinant of a diagonal matrix with diagonal entries d_1, d_2, …, d_n?

• Sum d_1 + … + d_n
• Product d_1 * d_2 * … * d_n
• d_1^n
• Zero if any d_i is nonzero

Correct Answer: Product d_1 * d_2 * … * d_n

Q19. How does the determinant relate to volume when a linear transformation represented by A acts on a region?

• Determinant is the new volume divided by n
• Absolute value of determinant gives the scaling factor of volume
• Determinant equals surface area scaling
• Determinant has no geometric interpretation

Correct Answer: Absolute value of determinant gives the scaling factor of volume

Q20. Using Cramer’s rule to solve Ax = b requires which determinant condition?

• det(A) = 0
• det(A) ≠ 0
• det(A) must be integer
• det(A) must be negative

Correct Answer: det(A) ≠ 0

Q21. Which formula gives the inverse of A using the adjugate when det(A) ≠ 0?

• A^{-1} = det(A) * adj(A)
• A^{-1} = adj(A) / det(A)
• A^{-1} = adj(A) * det(A)^2
• A^{-1} = adj(A) – det(A)

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

Q22. If det(A) = 5 and det(B) = 3 for 3×3 matrices, what is det(AB^{-1})?

• 15
• 5/3
• 5 * B
• 3/5

Correct Answer: 5/3

Q23. Which elementary row operation corresponds to multiplying the determinant by a scalar k?

• Swapping two rows
• Multiplying a row by k
• Adding a multiple of one row to another row
• Replacing two rows by their sum

Correct Answer: Multiplying a row by k

Q24. For an n x n matrix A, what is det(A^k) for integer k ≥ 1?

• det(A)^k
• k * det(A)
• det(kA)
• det(A) / k

Correct Answer: det(A)^k

Q25. If a square matrix A has two proportional rows, what is det(A)?

• Nonzero and equal to product of proportions
• Zero
• Equal to trace(A)
• Equal to determinant of transpose only

Correct Answer: Zero

Q26. Which matrix has determinant equal to the sign of the corresponding permutation?

• Identity matrix
• Permutation matrix
• Diagonal matrix of ones
• Zero matrix

Correct Answer: Permutation matrix

Q27. For a real orthogonal matrix Q (Q^T Q = I), what can be said about det(Q)?

• det(Q) is 0
• det(Q) is ±1
• det(Q) is always 1
• det(Q) is always negative

Correct Answer: det(Q) is ±1

Q28. Which property ensures det(P^{-1}AP) = det(A) for invertible P?

• Determinant is additive
• Determinant is invariant under similarity transformations
• Determinant depends on basis
• Determinant equals trace for similar matrices

Correct Answer: Determinant is invariant under similarity transformations

Q29. What is the determinant of the zero matrix of size n x n?

• n
• 0
• 1
• Undefined

Correct Answer: 0

Q30. Which statement about multilinearity of the determinant is correct?

• Determinant is linear in each row separately while other rows fixed
• Determinant is linear in all rows together only
• Determinant is nonlinear in every row
• Determinant is linear in columns but not rows

Correct Answer: Determinant is linear in each row separately while other rows fixed

Q31. The Sarrus rule is a mnemonic for computing determinants of which size matrix?

• 2×2
• 3×3
• 4×4
• Any n x n matrix

Correct Answer: 3×3

Q32. If A has two identical columns, what is det(A)?

• Equal to product of diagonal
• Zero
• Equal to sum of column entries
• Equal to determinant of transpose only

Correct Answer: Zero

Q33. In pharmacokinetic compartment models represented by linear systems, why are determinants important?

• They determine drug color
• They help assess system invertibility and solution existence
• They are unrelated to models
• They fix dosing units

Correct Answer: They help assess system invertibility and solution existence

Q34. What is the determinant of a scalar matrix kI_n?

• k
• k^n
• n*k
• 0

Correct Answer: k^n

Q35. If det(A) = -2 for a 2×2 matrix A, what is det(-A)?

• -2
• 2
• (-1)^2 * (-2) = -2
• (-1)^n det(A) = 2

Correct Answer: 2

Q36. For a skew-symmetric matrix S of odd order n, what is det(S)?

• Nonzero real
• Zero
• Always positive
• Equal to trace(S)

Correct Answer: Zero

Q37. Which of these operations does NOT change the value of the determinant?

• Swapping two rows
• Multiplying a row by a nonzero scalar
• Adding a multiple of one row to another
• Scaling the entire matrix by k

Correct Answer: Adding a multiple of one row to another

Q38. What is det(A) if A has a row of all zeros?

• Zero
• One
• Equal to product of other rows
• Undefined

Correct Answer: Zero

Q39. How is the determinant used in change of variables for multiple integrals (Jacobian)?

• It gives the rotation angle
• Its absolute value scales differential volume elements
• It is not used in change of variables
• It determines the integration limits only

Correct Answer: Its absolute value scales differential volume elements

Q40. Which of the following equals det(A) when computing determinant by expansion along column j?

• Sum over i of a_{ij} * C_{ij}
• Product over i of a_{ij}
• Sum of minors M_{ij} only
• Sum over i of a_{ji} * C_{ji}

Correct Answer: Sum over i of a_{ij} * C_{ij}

Q41. If det(A) = 4 for 4×4 matrix A, what is det(2A)?

• 8
• 16
• 32
• 4

Correct Answer: 32

Q42. The determinant of a matrix equals zero implies which of the following about its columns?

• They are orthogonal
• They are linearly dependent
• They form a basis of R^n
• They are all zero

Correct Answer: They are linearly dependent

Q43. What is the determinant of a 3×3 matrix with rows [1,2,3], [4,5,6], [7,8,9]?

• 0
• 1
• -3
• 45

Correct Answer: 0

Q44. If A is n x n and P is invertible, which identity holds true?

• det(PAP^{-1}) = det(A)
• det(PAP^{-1}) = det(P) + det(A) – det(P^{-1})
• det(PAP^{-1}) = det(P) * det(A) * det(P^{-1})^2
• det(PAP^{-1}) = 0

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

Q45. For a 3×3 matrix A, which is a quick test for singularity?

• Check if any diagonal entry is zero
• Compute determinant; if zero, singular
• Check if sum of entries is zero
• Check if trace equals determinant

Correct Answer: Compute determinant; if zero, singular

Q46. What is the determinant of the matrix [[2,0,0],[0,3,0],[0,0,4]]?

• 9
• 24
• 0
• 2+3+4

Correct Answer: 24

Q47. If det(A) = -5, what is det(-I * A) for a 3×3 A (where I is identity)?

• -5
• -125
• -5 * (-1)^3 = 5
• (-1)^3 * det(A) = 5

Correct Answer: 5

Q48. Which of the following elementary matrices has determinant equal to 1?

• Matrix that swaps two rows
• Matrix that multiplies a row by nonunit scalar
• Matrix that adds a multiple of one row to another
• Zero matrix

Correct Answer: Matrix that adds a multiple of one row to another

Q49. In practice, why might B. Pharm students use determinant properties in computations?

• To color code experiments
• To simplify solving linear compartmental models and evaluate invertibility quickly
• To determine molecular weight directly
• To change drug names

Correct Answer: To simplify solving linear compartmental models and evaluate invertibility quickly

Q50. Compute the determinant of [[1,0,2],[0,3,0],[4,0,5]].

• 1*3*5 – 0
• 3*(1*5 – 2*4) = 3*(5 – 8) = -9
• 0
• 1+3+5

Correct Answer: 3*(1*5 – 2*4) = 3*(5 – 8) = -9

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