Adjoint or adjugate of a square matrix MCQs With Answer

Understanding the adjoint (or adjugate) of a square matrix is essential for B. Pharm students studying pharmacy mathematics and linear algebra applications in pharmacokinetics. The adjoint, defined as the transpose of the cofactor matrix, ties together cofactors, minors, determinants and inverse matrices. Key properties—such as A·adj(A) = det(A) I and A⁻¹ = adj(A)/det(A) for non-singular matrices—help solve linear systems in drug modeling, formulation calculations and data fitting. This introduction gives clear definitions, computational steps and important identities to strengthen conceptual understanding and exam readiness. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the adjoint (adjugate) of a square matrix A?

• The matrix of minors of A
• The transpose of the cofactor matrix of A
• The inverse of A
• The determinant of A

Correct Answer: The transpose of the cofactor matrix of A

Q2. For a 2×2 matrix A = [[a, b], [c, d]], what is adj(A)?

• [[a, b], [c, d]]
• [[d, -b], [-c, a]]
• [[d, c], [b, a]]
• [[a, -c], [-b, d]]

Correct Answer: [[d, -b], [-c, a]]

Q3. Which relation between A and adj(A) holds for any n×n matrix A?

• A + adj(A) = det(A) I
• A · adj(A) = det(A) I
• adj(A) · A = I
• adj(A) = det(A) A

Correct Answer: A · adj(A) = det(A) I

Q4. If det(A) ≠ 0, what formula gives A^{-1} using adj(A)?

• A^{-1} = adj(A) · det(A)
• A^{-1} = adj(A) / det(A)
• A^{-1} = det(A) / adj(A)
• A^{-1} = adj(A) – det(A) I

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

Q5. The (i, j)-cofactor C_{ij} equals which of the following?

• (-1)^{i+j} times the minor M_{ij}
• Minor M_{ij} without sign
• Transpose of minor M_{ji}
• Determinant of A

Correct Answer: (-1)^{i+j} times the minor M_{ij}

Q6. How is adj(A) constructed from cofactors C_{ij}?

• Put cofactors in the same positions
• Take the transpose of the cofactor matrix
• Sum all cofactors into a single vector
• Multiply cofactors by det(A)

Correct Answer: Take the transpose of the cofactor matrix

Q7. For a 3×3 matrix A, computing adj(A) requires computing how many 2×2 minors?

• 3
• 6
• 9
• 12

Correct Answer: 9

Q8. Which identity is true for any two n×n matrices A and B (if defined)?

• adj(AB) = adj(A)adj(B)
• adj(AB) = adj(B)adj(A)
• adj(AB) = adj(A)+adj(B)
• adj(AB) = det(A)adj(B)

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

Q9. What is adj(I_n), where I_n is the n×n identity matrix?

• Zero matrix
• I_n
• n·I_n
• det(I_n)·I_n

Correct Answer: I_n

Q10. If A is singular (det(A) = 0) and rank(A) ≤ n−2, what is adj(A)?

• Invertible
• Zero matrix
• Identity matrix scaled
• Rank-one matrix

Correct Answer: Zero matrix

Q11. If A is n×n and det(A)=0 but rank(A)=n−1, then adj(A) typically has what rank?

• 0
• 1
• n−1
• n

Correct Answer: 1

Q12. How does adj(cA) relate to adj(A) for scalar c and n×n A?

• adj(cA) = c·adj(A)
• adj(cA) = c^{n} adj(A)
• adj(cA) = c^{n-1} adj(A)
• adj(cA) = adj(A)/c

Correct Answer: adj(cA) = c^{n-1} adj(A)

Q13. Which expression gives det(adj(A)) in terms of det(A) for n×n A?

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

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

Q14. For A 2×2 with entries a,b,c,d, what is det(adj(A)) equal to?

• (ad−bc)^2
• ad−bc
• (ad+bc)^2
• 0

Correct Answer: (ad−bc)^2

Q15. Which of the following is true about adj(A^T)?

• adj(A^T) = adj(A)
• adj(A^T) = adj(A)^T
• adj(A^T) = [adj(A)]^{-1}
• adj(A^T) = det(A) I

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

Q16. If A is invertible, what is adj(A^{-1}) equal to?

• det(A) A
• A
• det(A)^{-1} A
• adj(A)^{-1}

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

Q17. True or false: adj(A) is always invertible when A is invertible.

• True
• False
• Only when det(A)=1
• Only when A is diagonal

Correct Answer: True

Q18. What sign factor multiplies a minor to form its cofactor?

• (-1)^{i−j}
• (-1)^{i+j}
• (-1)^{ij}
• +1 always

Correct Answer: (-1)^{i+j}

Q19. In solving linear equations Ax = b using adj(A), what is required for the formula x = adj(A)b/det(A)?

• No requirement; works always
• det(A) = 0
• det(A) ≠ 0
• A must be diagonal

Correct Answer: det(A) ≠ 0

Q20. For a 3×3 matrix, cofactor C_{11} is computed from which submatrix?

• Remove row 1 and column 1
• Remove row 1 and column 2
• Remove row 2 and column 1
• Remove row 3 and column 3

Correct Answer: Remove row 1 and column 1

Q21. Which operation directly follows computing cofactors to obtain adj(A)?

• Add all cofactors
• Take the inverse of the cofactor matrix
• Transpose the cofactor matrix
• Multiply cofactors by det(A)

Correct Answer: Transpose the cofactor matrix

Q22. If A is 3×3 and adj(A) = 0 (zero matrix), what can be inferred about A?

• A is invertible
• rank(A) ≤ 1
• rank(A) ≤ 1 or ≤ n−2; specifically rank ≤ 1
• rank(A) ≤ n−2 (i.e., ≤ 1 for n=3)

Correct Answer: rank(A) ≤ n−2 (i.e., ≤ 1 for n=3)

Q23. Which of these is a practical application of adj(A) in pharmacy-related computations?

• Computing drug molecular weight
• Solving linear compartmental models
• Visualizing chemical structures
• Measuring pH directly

Correct Answer: Solving linear compartmental models

Q24. If adj(A) · A = det(A) I, what happens when det(A)=0?

• adj(A) must be inverse of A
• Product is zero matrix
• Product equals identity
• Equation is invalid

Correct Answer: Product is zero matrix

Q25. For a 1×1 matrix A = [a], what is adj(A)?

• [1]
• [a]
• [0]
• [a^{-1}]

Correct Answer: [1]

Q26. True or false: adj(A + B) = adj(A) + adj(B) for n×n matrices A,B.

• True
• False
• True only if A and B commute
• True only if det(A)=det(B)

Correct Answer: False

Q27. Which computational step is NOT part of computing adj(A) for a 3×3 matrix?

• Compute nine 2×2 minors
• Apply sign pattern (-1)^{i+j} to each minor
• Form cofactor matrix and transpose it
• Divide cofactor matrix by det(A)

Correct Answer: Divide cofactor matrix by det(A)

Q28. If A is diagonal with diagonal entries d1,…,dn, what is adj(A)?

• Diagonal with entries product of all but corresponding di
• Zero matrix
• Inverse of A
• Permutation of diagonal entries

Correct Answer: Diagonal with entries product of all but corresponding di

Q29. For a 3×3 matrix A, which of the following gives the (2,3)-cofactor location in adj(A)?

• Entry at row 2, column 3 of adj(A)
• Entry at row 3, column 2 of adj(A)
• Entry at row 2, column 3 of cofactor matrix before transpose
• Entry at row 3, column 2 of cofactor matrix before transpose

Correct Answer: Entry at row 3, column 2 of cofactor matrix before transpose

Q30. Which property helps to compute inverse quickly for 2×2 matrices using adjoint?

• adj(A) equals A
• A^{-1} = adj(A)/det(A) with simple adj(A) formula
• det(A) is always 1 for 2×2
• cofactor equals minor for 2×2

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

Q31. If A has integer entries and det(A)=±1, what can be said about adj(A)?

• adj(A) has integer entries
• adj(A) must be fractional
• adj(A) is zero
• adj(A) has determinant 0

Correct Answer: adj(A) has integer entries

Q32. True or false: adj(A) depends on the ordering of basis vectors chosen for A.

• True
• False
• True only in non-square matrices
• True only if A is singular

Correct Answer: False

Q33. When computing cofactors for a 4×4 matrix, each minor is a determinant of what size?

• 1×1
• 2×2
• 3×3
• 4×4

Correct Answer: 3×3

Q34. For n×n A, which identity connects adj(A) and A^{-1} when det(A) ≠ 0?

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

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

Q35. Which of the following is a quick check that adj(A) was computed correctly for 3×3 A?

• adj(A) is symmetric
• A · adj(A) equals det(A) I
• Sum of entries of adj(A) equals det(A)
• adj(A) has zero determinant

Correct Answer: A · adj(A) equals det(A) I

Q36. If A is 3×3 with det(A)=2, what is det(adj(A))?

• 2
• 4
• 8
• 16

Correct Answer: 4

Q37. Which technique reduces computation for adj(A) when A has a row of zeros except one entry?

• Use expansion by that row to simplify cofactors
• Compute full set of minors anyway
• Swap rows repeatedly
• Use LU decomposition only

Correct Answer: Use expansion by that row to simplify cofactors

Q38. For a matrix used in a pharmacokinetic compartment model, why is adj(A) useful?

• Helps to compute inverse and solve linear rates
• Measures drug potency directly
• Generates graphical plots of concentration
• Replaces experimental data

Correct Answer: Helps to compute inverse and solve linear rates

Q39. If A is 4×4 and adj(A) ≠ 0 but det(A)=0, which is likely true?

• rank(A) = 4
• rank(A) = 3
• rank(A) = 2
• rank(A) = 0

Correct Answer: rank(A) = 3

Q40. True or false: adj(A) is linear in the entries of A.

• True
• False
• True only for 2×2 matrices
• True only if det(A)=1

Correct Answer: False

Q41. Which of the following describes a minor M_{ij}?

• Determinant of matrix formed by deleting row i and column j
• Cofactor multiplied by (-1)^{i+j}
• Entry at position (i,j) in A
• Transpose of cofactor

Correct Answer: Determinant of matrix formed by deleting row i and column j

Q42. For a 2×2 A with det(A)=0, what is adj(A)?

• Always zero
• Proportional to A
• May be non-zero rank-one matrix
• Equal to identity

Correct Answer: May be non-zero rank-one matrix

Q43. Which of the following is a correct scalar identity useful in proofs with adj(A)?

• adj(A)A = A + det(A)I
• Trace(adj(A)) = det(A)
• adj(A)A = det(A)I
• adj(A) + A = det(A)I

Correct Answer: adj(A)A = det(A)I

Q44. If columns of A are linearly dependent, what effect does this have on adj(A)?

• adj(A) is guaranteed invertible
• adj(A) may be zero or rank-one
• adj(A) equals transpose of A
• adj(A) has full rank

Correct Answer: adj(A) may be zero or rank-one

Q45. In symbolic computations, why might adj(A) be preferred over computing A^{-1} directly?

• adj(A) avoids division by det(A) when det=0
• adj(A) gives a numerator useful in Cramer-like formulas
• adj(A) requires fewer operations always
• adj(A) is always diagonalizable

Correct Answer: adj(A) gives a numerator useful in Cramer-like formulas

Q46. Which of the following is true for polynomial matrices regarding adjoint?

• adj(A(x)) is polynomial in x if A(x) is polynomial
• adj(A(x)) is always constant
• adj(A(x)) cannot be computed symbolically
• adj(A(x)) equals derivative of det(A(x))

Correct Answer: adj(A(x)) is polynomial in x if A(x) is polynomial

Q47. If two rows of A are identical, what is adj(A)?

• Identity matrix
• Zero matrix
• Singular but non-zero
• Equal to A

Correct Answer: Zero matrix

Q48. Which step reduces numerical error when computing adj(A) for nearly singular matrices?

• Compute cofactors directly with high-precision arithmetic
• Use integer rounding
• Swap rows arbitrarily
• Ignore small cofactors

Correct Answer: Compute cofactors directly with high-precision arithmetic

Q49. For a 3×3 matrix A used to transform concentration vectors, computing adj(A) helps primarily to:

• Scale units of concentration
• Solve for original concentrations via inverse transform
• Plot time series
• Normalize probabilities

Correct Answer: Solve for original concentrations via inverse transform

Q50. Which identity correctly relates adj(A) to eigenvalues λ_i of A (assuming A diagonalizable)?

• Eigenvalues of adj(A) are λ_i
• Eigenvalues of adj(A) are product of all λ_j except λ_i
• Eigenvalues of adj(A) are 1/λ_i
• adj(A) has no relation to eigenvalues

Correct Answer: Eigenvalues of adj(A) are product of all λ_j except λ_i

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