Minors and Co-Factors MCQs With Answer

Introduction:

Minors and cofactors are fundamental matrix concepts used in determinants, inverse matrices, Cramer’s rule and systems of linear equations—skills essential for B.Pharm students handling pharmacokinetic models, formulation stoichiometry and biostatistics. This concise guide reinforces the definitions of a minor, cofactor, adjugate (adjoint) matrix, Laplace expansion, and how to compute inverses using cofactors. Emphasis is on practical computation, determinant properties, rank implications and real-life applications in drug-dose calculations and compartmental models. Clear examples and targeted practice enhance problem-solving for exams and labs. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the definition of the minor of an element a_ij in a matrix?

• The determinant of the submatrix formed by deleting row i and column j
• The algebraic complement of a_ij with sign factor
• The element obtained by transposing the matrix
• The sum of elements in row i and column j

Correct Answer: The determinant of the submatrix formed by deleting row i and column j

Q2. How is the cofactor C_ij related to the minor M_ij?

• C_ij = (-1)^(i+j) * M_ij
• C_ij = M_ij / (-1)^(i+j)
• C_ij = M_ij + (-1)^(i+j)
• C_ij = determinant of full matrix minus M_ij

Correct Answer: C_ij = (-1)^(i+j) * M_ij

Q3. Which method uses minors and cofactors to compute the inverse of a non-singular matrix?

• Adjugate (adjoint) method
• Gaussian elimination without pivoting
• LU decomposition only
• Power method

Correct Answer: Adjugate (adjoint) method

Q4. For a 2×2 matrix [[a,b],[c,d]], what is the minor of element a (position 1,1)?

• d
• b
• a

Correct Answer: d

Q5. For the same 2×2 matrix, what is the cofactor of element b (position 1,2)?

• -c
• c
• -d
• d

Correct Answer: -c

Q6. The adjugate (adjoint) matrix is defined as:

• The transpose of the cofactor matrix
• The inverse of the determinant
• The matrix of minors without sign changes
• The square of the original matrix

Correct Answer: The transpose of the cofactor matrix

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

• A is singular and non-invertible
• A must be diagonal
• A has an inverse equal to adj(A)
• A has full rank

Correct Answer: A is singular and non-invertible

Q8. Which expansion technique uses minors and cofactors to calculate a determinant?

• Laplace expansion
• Taylor series expansion
• Fourier expansion
• Binomial expansion

Correct Answer: Laplace expansion

Q9. What is the sign pattern for cofactors in a 3×3 matrix?

• + – +; – + -; + – +
• + + +; + + +; + + +
• – + -; + – +; – + –
• – – -; – – -; – – –

Correct Answer: + – +; – + -; + – +

Q10. In computing the inverse A^-1 via adj(A), the formula is:

• A^-1 = adj(A) / det(A)
• A^-1 = det(A) / adj(A)
• A^-1 = adj(A) * det(A)
• A^-1 = adj(A) + det(A)

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

Q11. The minor of element at row 2 column 3 is computed by:

• Deleting row 2 and column 3 then taking determinant of remaining matrix
• Multiplying row 2 and column 3 elements
• Summing all elements in row 2 and column 3
• Transposing the matrix and taking that element

Correct Answer: Deleting row 2 and column 3 then taking determinant of remaining matrix

Q12. In pharmacokinetic modeling, matrices and their inverses (via cofactors) help solve:

• Systems of linear compartment equations for drug distribution
• Protein sequence alignment
• pH titration curves directly without equations
• Qualitative solubility predictions only

Correct Answer: Systems of linear compartment equations for drug distribution

Q13. Which property relates cofactors to the determinant of A when multiplied by A?

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

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

Q14. If a matrix has rank r, what is the largest order of a non-zero minor?

• r
• 1
• n (matrix order)
• r+1

Correct Answer: r

Q15. A principal minor is obtained by deleting which rows and columns?

• The same set of row and column indices
• Different rows and columns randomly
• Only the first row and last column
• Only diagonal elements

Correct Answer: The same set of row and column indices

Q16. When expanding determinant along a row containing many zeros, best practice is to:

• Expand along that row to simplify computation
• Always expand along first row regardless
• Convert to scalar via trace first
• Compute inverse instead

Correct Answer: Expand along that row to simplify computation

Q17. For a 3×3 matrix, cofactor expansion along the first column yields the determinant as:

• a_11 C_11 + a_21 C_21 + a_31 C_31
• a_11 M_11 + a_21 M_21 + a_31 M_31
• sum of all minors only
• product of diagonal elements

Correct Answer: a_11 C_11 + a_21 C_21 + a_31 C_31

Q18. Which of the following statements is true about cofactor matrix?

• Its transpose is the adjugate matrix
• It is always symmetric
• It equals the inverse for singular matrices
• It is the identity matrix for all invertible matrices

Correct Answer: Its transpose is the adjugate matrix

Q19. The algebraic cofactor is sometimes called:

• Signed minor
• Principal element
• Diagonal complement
• Orthogonal factor

Correct Answer: Signed minor

Q20. If you interchange two rows of a matrix, how does determinant change?

• Sign of determinant changes (multiplied by -1)
• Determinant doubles
• Determinant remains same
• Determinant becomes zero always

Correct Answer: Sign of determinant changes (multiplied by -1)

Q21. For matrix A, what does a zero cofactor for every element in a row indicate?

• Determinant is zero (row is linearly dependent)
• Matrix is diagonalizable only
• Matrix has full rank
• Matrix is invertible with determinant one

Correct Answer: Determinant is zero (row is linearly dependent)

Q22. How does multiplying a row by scalar k affect minors that include that row?

• Those minors are multiplied by k
• Those minors remain unchanged
• All minors are multiplied by k squared
• Minors become negative

Correct Answer: Those minors are multiplied by k

Q23. What is the minor M_12 of matrix [[1,2,3],[4,5,6],[7,8,9]]?

• det([[4,6],[7,9]]) = (4*9 – 6*7) = -6
• det([[1,3],[7,9]]) = -4
• det([[1,2],[7,8]]) = -6
• det([[4,5],[7,8]]) = -3

Correct Answer: det([[4,6],[7,9]]) = (4*9 – 6*7) = -6

Q24. What is the cofactor C_12 for the same matrix in Q23?

• C_12 = (-1)^(1+2) * M_12 = -(-6) = 6
• C_12 = (-1)^(1+2) * M_12 = -6
• C_12 = 0
• C_12 = (-1)^(1+2) * M_12 = -12

Correct Answer: C_12 = (-1)^(1+2) * M_12 = -(-6) = 6

Q25. Which of the following helps compute unknowns in linear dose systems using determinants?

• Cramer’s rule
• Euler’s method
• Newton’s cooling law
• Beer-Lambert law

Correct Answer: Cramer’s rule

Q26. In Cramer’s rule, what role do minors and cofactors play?

• They help compute determinants of coefficient matrices for numerator replacements
• They are irrelevant to Cramer’s rule
• They only scale the solution after inversion
• They replace unknowns with zeros

Correct Answer: They help compute determinants of coefficient matrices for numerator replacements

Q27. If the determinant of a 3×3 matrix A is 5, what is det(2A)?

• 2^3 * det(A) = 8 * 5 = 40
• 2 * 5 = 10
• det(A) unchanged = 5
• det(2A) = det(A)^2 = 25

Correct Answer: 2^3 * det(A) = 8 * 5 = 40

Q28. The minor matrix (matrix of minors) differs from the cofactor matrix by:

• Sign pattern multiplication elementwise
• Transposition only
• Division by determinant
• Scalar addition of identity

Correct Answer: Sign pattern multiplication elementwise

Q29. For solving simultaneous equations in drug formulation, why are determinants useful?

• They provide solution existence (non-zero det) and allow explicit formulas
• They linearize nonlinear pharmacokinetics automatically
• They estimate variance in bioassays directly
• They replace experimental data

Correct Answer: They provide solution existence (non-zero det) and allow explicit formulas

Q30. If A is 3×3 and one row is scalar multiple of another, what can be said about all 2×2 minors containing those two rows?

• They are zero
• They equal determinant of A
• They equal one
• They are non-zero necessarily

Correct Answer: They are zero

Q31. Which statement is true about the cofactor matrix of an identity matrix I_n?

• Cofactor matrix = I_n (since minors are determinants of smaller identity submatrices with sign)
• Cofactor matrix = zero matrix
• Cofactor matrix = 2*I_n
• Cofactor matrix = diagonal of zeros

Correct Answer: Cofactor matrix = I_n (since minors are determinants of smaller identity submatrices with sign)

Q32. What is the effect on the determinant if two columns are identical?

• Determinant is zero
• Determinant doubles
• Determinant remains unchanged
• Determinant becomes negative inverse

Correct Answer: Determinant is zero

Q33. Which minor order identifies the rank of a matrix?

• The highest order non-zero minor equals the rank
• Only first order minors determine rank
• Sum of all minors gives rank
• Minors do not relate to rank

Correct Answer: The highest order non-zero minor equals the rank

Q34. In a 3×3, if all 2×2 minors are zero but determinant is zero, rank is likely:

• 1 or 0 depending on 1×1 minors
• 3
• 2
• Undefined

Correct Answer: 1 or 0 depending on 1×1 minors

Q35. Which computational step is required to form adj(A) for a 3×3?

• Compute all 3×3 cofactors then transpose the cofactor matrix
• Only compute determinant
• Square the cofactor matrix
• Invert each element individually

Correct Answer: Compute all 3×3 cofactors then transpose the cofactor matrix

Q36. For 4×4 matrices, minors used in expansion are of order:

• 3 when expanding along one row/column
• 4 only
• 1 only
• 5

Correct Answer: 3 when expanding along one row/column

Q37. If A*X = B and det(A) ≠ 0, how can X be found using cofactors?

• X = A^-1 * B where A^-1 = adj(A)/det(A)
• X equals adj(A) without division
• X cannot be found by cofactors
• X = det(A) * adj(A) * B

Correct Answer: X = A^-1 * B where A^-1 = adj(A)/det(A)

Q38. Which of the following is NOT true about minors?

• Minors are not affected by adding a multiple of one row to another row
• Minors are determinants of submatrices
• Minors always equal cofactors
• Minors change sign when corresponding cofactor sign factor applied

Correct Answer: Minors always equal cofactors

Q39. In biochemical network linearization, why might one compute cofactors?

• To invert Jacobian matrices when assessing stability near equilibrium
• To sequence proteins
• To amplify DNA samples
• To measure pH directly

Correct Answer: To invert Jacobian matrices when assessing stability near equilibrium

Q40. Which formula gives determinant by expansion along i-th row?

• det(A) = sum_j a_ij * C_ij
• det(A) = product_j a_ij
• det(A) = sum_j M_ij only
• det(A) = trace(A)

Correct Answer: det(A) = sum_j a_ij * C_ij

Q41. For matrix [[2,0,0],[0,3,0],[0,0,4]], what is adj(A)?

• diag(12,8,6) (transpose of cofactor matrix yields these diagonal entries)
• diag(2,3,4)
• Zero matrix
• Matrix with all entries 1

Correct Answer: diag(12,8,6) (transpose of cofactor matrix yields these diagonal entries)

Q42. If one computes cofactor matrix and then multiplies by A, resulting product equals:

• det(A) times identity matrix
• zero matrix always
• adj(A) only
• A inverse

Correct Answer: det(A) times identity matrix

Q43. Which is an advantage of using cofactor/adjoint method for inverse?

• Exact symbolic inverse useful for algebraic analysis
• Always faster than numerical methods for large matrices
• Requires no determinant computation
• Avoids calculation of minors

Correct Answer: Exact symbolic inverse useful for algebraic analysis

Q44. What is a drawback of adjugate method for large matrices?

• Computationally expensive due to many minors and cofactors
• Gives approximate results only
• Cannot be used if determinant non-zero
• Destroys matrix sparsity always

Correct Answer: Computationally expensive due to many minors and cofactors

Q45. For a triangular matrix, minors used for determinant computation are:

• Simplified since determinant equals product of diagonal entries
• Always zero
• Equal to sum of off-diagonal elements
• Not defined

Correct Answer: Simplified since determinant equals product of diagonal entries

Q46. A cofactor expansion along a column is equivalent to expansion along a row after:

• Appropriate transposition of matrix
• Multiplying matrix by two
• Subtracting identity
• Adding a zero row

Correct Answer: Appropriate transposition of matrix

Q47. If adj(A) is computed and det(A)=1, then A^-1 equals:

• adj(A)
• det(A) * adj(A)
• Zero matrix
• Transpose of A

Correct Answer: adj(A)

Q48. Which calculation directly uses minors when assessing matrix rank in practice?

• Checking existence of non-zero k-order minors for descending k
• Computing eigenvalues only
• Performing Fourier transform
• Measuring sample concentration

Correct Answer: Checking existence of non-zero k-order minors for descending k

Q49. In numerical practice for B.Pharm problems, when is adjoint/cofactor method most appropriate?

• For small matrices or symbolic derivations where exact inverse is needed
• For very large sparse systems normally
• When determinant equals zero
• When real-time computation is required for large simulations

Correct Answer: For small matrices or symbolic derivations where exact inverse is needed

Q50. Which is a correct practical tip when computing cofactors by hand?

• Choose expansion along row/column with most zeros to reduce work
• Always expand along the last row to avoid sign errors
• Compute all minors of full order first
• Avoid computing determinant

Correct Answer: Choose expansion along row/column with most zeros to reduce work

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