Transpose of a matrix MCQs With Answer

In B.Pharm courses, understanding the Transpose of a matrix is essential for calculations in drug dosage modeling, pharmacokinetics, and chemoinformatics. This concise guide on “Transpose of a matrix MCQs With Answer” covers definitions, properties, notation (A^T), symmetry, and operations that B.Pharm students encounter in pharmaceutical problem solving. Questions focus on row-column interchange, effects on determinants and ranks, symmetric and skew-symmetric matrices, and applications in linear systems used in formulation and QC data analysis. Practical examples and step-by-step explanations improve problem-solving speed. Clear diagrams, practice problems, and exam-style tips are included to help you master matrix transpose concepts quickly. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the transpose of a 2×3 matrix?

• A 3×2 matrix formed by interchanging rows and columns
• A 2×3 matrix with negated entries
• A diagonal matrix composed of original row sums
• A scalar equal to the sum of all elements

Correct Answer: A 3×2 matrix formed by interchanging rows and columns

Q2. If A is a 2×2 matrix with entries [[1,2],[3,4]], what is A^T?

• [[1,3],[2,4]]
• [[4,3],[2,1]]
• [[1,2],[3,4]]
• [[2,1],[4,3]]

Correct Answer: [[1,3],[2,4]]

Q3. Which property is always true for any matrices A and B where dimensions allow addition?

• (A + B)^T = A^T + B^T
• (A + B)^T = A^T – B^T
• (A + B)^T = B + A
• (A + B)^T = (A^T + B)^2

Correct Answer: (A + B)^T = A^T + B^T

Q4. For conformable matrices A (m×n) and B (n×p), what is (AB)^T?

• B^T A^T
• A^T B^T
• A B
• B A

Correct Answer: B^T A^T

Q5. If A is invertible, which identity holds involving transpose and inverse?

• (A^{-1})^T = (A^T)^{-1}
• (A^{-1})^T = A^T
• (A^T)^{-1} = A
• (A^{-1})^T = -A^{-1}

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

Q6. Which statement about determinant and transpose is correct for any square matrix A?

• det(A^T) = det(A)
• det(A^T) = -det(A)
• det(A^T) = 1/det(A)
• det(A^T) = (det(A))^2

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

Q7. What is the transpose of a row vector [a b c]?

• Column vector [a; b; c]
• Row vector [c b a]
• Scalar a+b+c
• Zero vector

Correct Answer: Column vector [a; b; c]

Q8. Which of the following matrices satisfies A^T = A?

• Symmetric matrix
• Skew-symmetric matrix
• Orthogonal matrix only
• Diagonal matrix only

Correct Answer: Symmetric matrix

Q9. A matrix A satisfies A^T = -A. What type of matrix is A?

• Skew-symmetric
• Symmetric
• Orthogonal
• Singular

Correct Answer: Skew-symmetric

Q10. For any matrix A, what is (A^T)^T equal to?

• A
• A^T
• Zero matrix
• 2A

Correct Answer: A

Q11. If A is m×n, what are the dimensions of A^T A?

• n×n
• m×m
• m×n
• n×m

Correct Answer: n×n

Q12. Why is A^T A important in least-squares fitting used in pharmacokinetic modeling?

• It forms normal equations to compute best-fit parameters
• It always gives the inverse of A
• It reduces matrix rank to one
• It diagonalizes A automatically

Correct Answer: It forms normal equations to compute best-fit parameters

Q13. If A is orthogonal, which relation between A and A^T holds?

• A^T = A^{-1}
• A^T = -A
• A^T = A
• A^T = 0

Correct Answer: A^T = A^{-1}

Q14. For real matrices, which equality involving trace holds?

• trace(A^T) = trace(A)
• trace(A^T) = -trace(A)
• trace(A^T) = 0
• trace(A^T) = (trace(A))^2

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

Q15. What happens to the rank of a matrix when you take its transpose?

• Rank remains the same
• Rank doubles
• Rank becomes zero
• Rank becomes negative

Correct Answer: Rank remains the same

Q16. If A is 3×3 and symmetric, which of the following is true about its eigenvalues (real matrix)?

• All eigenvalues are real and eigenvectors can be chosen orthogonal
• All eigenvalues are zero
• Eigenvalues are complex conjugate pairs only
• Eigenvalues are always positive

Correct Answer: All eigenvalues are real and eigenvectors can be chosen orthogonal

Q17. Given matrix A = [[0,1],[-1,0]], what is A^T?

• [[0,-1],[1,0]]
• [[0,1],[-1,0]]
• [[1,0],[0,-1]]
• [[-0, -1],[1, 0]]

Correct Answer: [[0,-1],[1,0]]

Q18. Which statement is true about transpose of a scalar multiple?

• (kA)^T = k A^T
• (kA)^T = A^T / k
• (kA)^T = -k A^T
• (kA)^T = (A^T)^k

Correct Answer: (kA)^T = k A^T

Q19. In data analysis for QC, why might you compute X^T X for a data matrix X?

• To obtain covariance-like matrix and normal equations
• To transpose each sample individually
• To compute element-wise reciprocals
• To reduce the number of variables

Correct Answer: To obtain covariance-like matrix and normal equations

Q20. If A is 2×3: [[1,2,3],[4,5,6]], what is A^T?

• [[1,4],[2,5],[3,6]]
• [[1,2,3],[4,5,6]]
• [[6,5,4],[3,2,1]]
• [[1,3],[2,4],[5,6]]

Correct Answer: [[1,4],[2,5],[3,6]]

Q21. Which of the following matrices is both symmetric and skew-symmetric?

• The zero matrix
• Any diagonal matrix with nonzero entries
• Any identity matrix
• Any invertible matrix

Correct Answer: The zero matrix

Q22. Which relation connects transpose and complex conjugate for complex matrices?

• Conjugate transpose (A^*) is transpose plus complex conjugate
• A^T equals complex conjugate always
• Transpose removes imaginary parts only
• Transpose equals inverse for complex matrices

Correct Answer: Conjugate transpose (A^*) is transpose plus complex conjugate

Q23. For matrices A and B of same size, what is true about (A – B)^T?

• (A – B)^T = A^T – B^T
• (A – B)^T = B – A
• (A – B)^T = A^T + B^T
• (A – B)^T = -(A^T – B^T)

Correct Answer: (A – B)^T = A^T – B^T

Q24. Which of the following is a direct application of matrix transpose in computational pharmacology?

• Forming normal equations for parameter estimation
• Eliminating measurement noise directly
• Increasing the size of data matrices arbitrarily
• Converting nonlinear models to linear form automatically

Correct Answer: Forming normal equations for parameter estimation

Q25. If A is 4×2, what are the dimensions of A^T?

• 2×4
• 4×2
• 4×4
• 2×2

Correct Answer: 2×4

Q26. Which operation preserves symmetry: if A is symmetric, which of these is also symmetric?

• A + A^T
• A – A^T
• A A^T when A is arbitrary
• A^T – A

Correct Answer: A + A^T

Q27. If A is skew-symmetric, what can be said about diagonal entries of A (real matrix)?

• All diagonal entries are zero
• All diagonal entries are positive
• All diagonal entries are equal to one
• Diagonal entries are arbitrary reals

Correct Answer: All diagonal entries are zero

Q28. Given A = [[2,0],[0,3]], what is A^T?

• [[2,0],[0,3]]
• [[0,2],[3,0]]
• [[3,0],[0,2]]
• [[-2,0],[0,-3]]

Correct Answer: [[2,0],[0,3]]

Q29. For matrices A (m×n) and B (m×n), which statement is true about inner products using transpose?

• x^T y is scalar representing dot product of vectors x and y
• A^T B is always scalar
• Transpose converts inner product to cross product
• x^T y equals zero only for identical vectors

Correct Answer: x^T y is scalar representing dot product of vectors x and y

Q30. Which of the following is not generally true for transpose?

• (AB)^T = A^T B^T
• (A + B)^T = A^T + B^T
• (A^T)^T = A
• (kA)^T = k A^T

Correct Answer: (AB)^T = A^T B^T

Q31. In practice, why is the transpose used when constructing covariance matrices from data matrix X (rows observations, columns variables)?

• Because X^T X gives sums of products across observations for variable pairs
• Because X^T always inverts X
• Because transpose reduces noise variance directly
• Because transpose converts categorical data to continuous

Correct Answer: Because X^T X gives sums of products across observations for variable pairs

Q32. Which matrix property ensures A^T = A^{-1} and columns form an orthonormal basis?

• Orthogonality
• Singularity
• Skew-symmetry
• Positivity

Correct Answer: Orthogonality

Q33. If A is 1×n, what is A^T A?

• A scalar equal to sum of squares of entries
• An n×n matrix
• A 1×1 zero matrix always
• A vector of length n

Correct Answer: A scalar equal to sum of squares of entries

Q34. For a square matrix A, which equality involving transpose and trace is always true?

• trace(A^T B) = trace(A B^T)
• trace(A^T) = -trace(A)
• trace(A^T B) = 0 for all B
• trace(A^T) = trace(A)^2

Correct Answer: trace(A^T B) = trace(A B^T)

Q35. If matrix A has linearly independent columns, what can be said about A^T A?

• A^T A is invertible (positive definite)
• A^T A is singular
• A^T A is skew-symmetric
• A^T A is always diagonal

Correct Answer: A^T A is invertible (positive definite)

Q36. Which best describes the transpose of a product of three matrices ABC?

• (ABC)^T = C^T B^T A^T
• (ABC)^T = A^T B^T C^T
• (ABC)^T = B^T A^T C^T
• (ABC)^T = C A B

Correct Answer: (ABC)^T = C^T B^T A^T

Q37. In matrix notation, converting a system of linear equations Ax = b to normal equations involves which transpose operation?

• A^T A x = A^T b
• A A^T x = b
• x = A^T b
• A^T = b

Correct Answer: A^T A x = A^T b

Q38. Which of the following is true for a real skew-symmetric matrix S?

• S^T = -S
• S^T = S
• S^T = S^{-1}
• S^T = 0

Correct Answer: S^T = -S

Q39. If A is 3×3 with rank 2, what is rank(A^T)?

• 2
• 3
• 1
• 0

Correct Answer: 2

Q40. How does transpose help when implementing matrix operations in code for pharmacology data?

• It allows switching between row-major and column-major orientations to match algorithms
• It increases numerical precision automatically
• It removes outliers from the dataset
• It always speeds up computation regardless of context

Correct Answer: It allows switching between row-major and column-major orientations to match algorithms

Q41. If A is diagonal, what is A^T?

• Same as A
• Zero matrix
• Negative of A
• Transpose cannot be defined

Correct Answer: Same as A

Q42. For rectangular matrices, which statement is true?

• Transpose changes m×n to n×m but preserves linear relations among rows and columns
• Transpose always makes the matrix square
• Transpose inverts the matrix
• Transpose changes entries to their reciprocals

Correct Answer: Transpose changes m×n to n×m but preserves linear relations among rows and columns

Q43. If A is 2×2 and A^T = A^{-1}, what is determinant of A?

• ±1
• 0
• 2
• Any real number

Correct Answer: ±1

Q44. What is the transpose of the identity matrix I_n?

• I_n
• Zero matrix
• -I_n
• Matrix of ones

Correct Answer: I_n

Q45. In chemical kinetics modeling, which matrix operation often uses transpose to combine experimental design matrices?

• Forming X^T X for parameter estimation
• Element-wise exponentiation
• Computing cross products of scalars only
• Applying nonlinear activation directly

Correct Answer: Forming X^T X for parameter estimation

Q46. Which condition ensures A^T A is symmetric?

• Always for any A
• Only if A is square
• Only if A is symmetric
• Only if A is invertible

Correct Answer: Always for any A

Q47. If v is a column vector, what is v^T v?

• A scalar equal to squared norm of v
• A column vector of same size
• A matrix of zeros
• Undefined

Correct Answer: A scalar equal to squared norm of v

Q48. Which of the following explains why A and A^T have the same characteristic polynomial (for square A)?

• They are similar via permutation of basis, so eigenvalues match
• Transpose changes eigenvalues to their negatives
• Characteristic polynomial of A^T is always zero
• They never have same characteristic polynomial unless diagonal

Correct Answer: They are similar via permutation of basis, so eigenvalues match

Q49. For a 2×2 matrix A = [[a,b],[c,d]], which expression equals A^T?

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

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

Q50. In experimental design matrices used in pharmaceutics, why is transpose used before computing parameter covariance?

• Because covariance of estimates often involves (X^T X)^{-1}, requiring transpose
• Because transpose removes collinearity automatically
• Because transpose amplifies measurement error useful for detection
• Because transpose reduces matrix size to 1×1

Correct Answer: Because covariance of estimates often involves (X^T X)^{-1}, requiring transpose

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