Roots of a square matrix MCQs With Answer

Roots of a square matrix MCQs With Answer

Understanding the roots of a square matrix (matrix square root) is essential for B. Pharm students who study mathematical modeling, pharmacokinetics, and numerical methods. This concise guide explains when a square matrix A admits a matrix X with X² = A, highlights key ideas like eigenvalues, diagonalization, spectral decomposition, positive definite matrices, and numerical computation methods, and connects these ideas to practical uses such as parameter estimation and stability analysis. Clear definitions, existence and uniqueness conditions, and computation techniques are emphasized to build strong analytical skills. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is a square root of a square matrix A?

• A matrix X such that X + X = A
• A matrix X such that X² = A
• A matrix X such that AX = XA
• A matrix X such that X³ = A

Correct Answer: A matrix X such that X² = A

Q2. If A is diagonalizable as A = SDS⁻¹, where D is diagonal, how can a square root of A be constructed (when possible)?

• S D² S⁻¹
• S diag(√d_i) S⁻¹ where d_i are diagonal entries of D
• S D^(1/3) S⁻¹
• S diag(log d_i) S⁻¹

Correct Answer: S diag(√d_i) S⁻¹ where d_i are diagonal entries of D

Q3. Which class of matrices always has a unique symmetric positive definite square root?

• Any singular matrix
• Any orthogonal matrix
• Any real symmetric positive definite matrix
• Any skew-symmetric matrix

Correct Answer: Any real symmetric positive definite matrix

Q4. For a matrix X satisfying X² = A, what is the relationship between eigenvalues μ of X and eigenvalues λ of A?

• μ = λ²
• μ² = λ
• μ + μ = λ
• μ = log λ

Correct Answer: μ² = λ

Q5. Which statement is true about the zero matrix 0 (all zeros)?

• The only square root of 0 is the zero matrix
• 0 has no square roots
• 0 can have many nilpotent square roots besides the zero matrix
• Every matrix is a square root of 0

Correct Answer: 0 can have many nilpotent square roots besides the zero matrix

Q6. If A is invertible and X² = A, what can be said about X?

• X must be singular
• X is necessarily invertible
• X must be orthogonal
• X must be diagonalizable

Correct Answer: X is necessarily invertible

Q7. How does determinant behave under matrix square roots?

• det(X) = det(A)
• det(X)² = det(A)
• det(X) + det(X) = det(A)
• det(X) = 1/det(A)

Correct Answer: det(X)² = det(A)

Q8. Which method is commonly used to compute a matrix square root numerically?

• Gaussian elimination with partial pivoting
• Newton iteration (Denman–Beavers or similar)
• Simple elementwise square roots of entries
• LU decomposition without pivoting

Correct Answer: Newton iteration (Denman–Beavers or similar)

Q9. If A is a 2×2 diagonal matrix with positive diagonal entries a and b, which is a square root of A?

• Any matrix with entries ±√a and ±√b placed arbitrarily
• diag(√a, √b)
• diag(a², b²)
• Matrix with off-diagonal terms only

Correct Answer: diag(√a, √b)

Q10. Is the matrix square root always unique for a given matrix A?

• Yes, always unique
• No, never unique
• Not in general — uniqueness holds under extra conditions like positive definiteness
• Unique only for singular matrices

Correct Answer: Not in general — uniqueness holds under extra conditions like positive definiteness

Q11. For a real symmetric matrix A with a negative eigenvalue, which statement is correct about real symmetric square roots?

• A real symmetric square root exists and is positive definite
• A real symmetric square root cannot exist for A
• A real symmetric square root exists only if A is diagonalizable
• A real symmetric square root exists and is unique

Correct Answer: A real symmetric square root cannot exist for A

Q12. Which property is preserved by the principal square root of a Hermitian positive definite matrix?

• Skew-symmetry
• Hermitian positive definiteness
• Nilpotency
• Orthogonality

Correct Answer: Hermitian positive definiteness

Q13. If X is a square root of A and Y is similar to X (Y = SXS⁻¹), what is Y²?

• Y² = S A S⁻¹
• Y² = A
• Y² = S X² S
• Y² = X²

Correct Answer: Y² = S A S⁻¹

Q14. Can a non-diagonalizable matrix A have a square root?

• No, non-diagonalizable matrices never have square roots
• Yes, some non-diagonalizable matrices can have square roots
• Only if they are singular
• Only in odd dimensions

Correct Answer: Yes, some non-diagonalizable matrices can have square roots

Q15. For a diagonal matrix D with entries that include zero and positive numbers, which is true about square roots?

• D cannot have any square root if it contains zero
• Square roots exist and may be chosen by taking square roots of diagonal entries, choosing zero for zero entries
• Square roots must introduce off-diagonal terms
• Square roots exist only over complex numbers

Correct Answer: Square roots exist and may be chosen by taking square roots of diagonal entries, choosing zero for zero entries

Q16. If A is orthogonal (A⁻¹ = Aᵀ), does A always have a real square root that is also orthogonal?

• Yes, every orthogonal matrix has an orthogonal square root
• No, not every orthogonal matrix has an orthogonal square root
• Only identity has an orthogonal square root
• Orthogonal matrices cannot have square roots

Correct Answer: No, not every orthogonal matrix has an orthogonal square root

Q17. Which of these statements about eigenvalues is necessary (but not sufficient) for a real matrix A to have a real square root?

• All eigenvalues of A must be positive real numbers
• All eigenvalues of A must have square roots in the field considered (real or complex)
• All eigenvalues must be zero
• Eigenvalues must sum to 1

Correct Answer: All eigenvalues of A must have square roots in the field considered (real or complex)

Q18. In pharmacokinetic modeling, why might matrix square roots be useful?

• To convert compartment models into scalar equations only
• To help compute matrix exponentials, covariance square roots, or apply transformations in multivariate models
• They are not useful in pharmacokinetics
• To guarantee linear independence of parameters

Correct Answer: To help compute matrix exponentials, covariance square roots, or apply transformations in multivariate models

Q19. Which is a correct statement about computing a matrix square root by spectral decomposition for symmetric A?

• A = SΛS⁻¹ and sqrt(A) = SΛS⁻¹
• A = SΛSᵀ and sqrt(A) = SΛ^(1/2)Sᵀ
• A = SΛSᵀ and sqrt(A) = S²ΛS⁻²
• Spectral decomposition cannot be used for symmetric matrices

Correct Answer: A = SΛSᵀ and sqrt(A) = SΛ^(1/2)Sᵀ

Q20. Which matrix definitely does not have a real square root that is symmetric positive definite?

• A symmetric matrix with strictly positive eigenvalues
• An identity matrix
• A symmetric matrix with a negative eigenvalue
• A diagonal matrix with positive entries

Correct Answer: A symmetric matrix with a negative eigenvalue

Q21. If X is a square root of A, is -X also a square root of A?

• No, never
• Yes, because (-X)² = X² = A
• Only if A is singular
• Only if X is symmetric

Correct Answer: Yes, because (-X)² = X² = A

Q22. Which requirement ensures the uniqueness of the principal square root for complex matrices?

• A must be diagonalizable
• A must have no eigenvalues on the negative real axis and be invertible (or be positive definite in Hermitian case)
• A must be nilpotent
• A must be singular

Correct Answer: A must have no eigenvalues on the negative real axis and be invertible (or be positive definite in Hermitian case)

Q23. The principal square root of a positive semidefinite matrix is:

• Not defined
• Also positive semidefinite
• Always invertible
• Always skew-symmetric

Correct Answer: Also positive semidefinite

Q24. Which is a necessary condition for a real matrix A to have a real square root X with X also real?

• All entries of A must be nonnegative
• All complex eigenvalues of A must occur in conjugate pairs so real X can exist
• A must be diagonal
• Trace(A) must be positive

Correct Answer: All complex eigenvalues of A must occur in conjugate pairs so real X can exist

Q25. For a 2×2 rotation matrix R(θ), does there exist a real matrix X such that X² = R(θ)?

• Yes, for many angles θ (e.g., half-angle rotation)
• No, rotation matrices never have square roots
• Only when θ = 0
• Only if R is diagonal

Correct Answer: Yes, for many angles θ (e.g., half-angle rotation)

Q26. If A has eigenvalue λ = 0 with an odd-sized Jordan block, what difficulty can arise for square roots?

• No difficulty; square roots always exist
• There may be obstructions to existence or complicated Jordan structure for any square root
• It guarantees uniqueness of square root
• It implies A is diagonalizable

Correct Answer: There may be obstructions to existence or complicated Jordan structure for any square root

Q27. What is the easiest way to compute a square root of a diagonalizable matrix in practice?

• Compute an SVD and take square root of singular values
• Diagonalize A = S D S⁻¹ and compute S diag(√d_i) S⁻¹
• Compute inverse and square it
• Take elementwise square roots of A entries

Correct Answer: Diagonalize A = S D S⁻¹ and compute S diag(√d_i) S⁻¹

Q28. Which statement is true about singular matrices and square roots?

• Singular matrices never have square roots
• Singular matrices may have square roots, including noninvertible ones
• Any singular matrix has infinitely many square roots
• Singular implies diagonalizable

Correct Answer: Singular matrices may have square roots, including noninvertible ones

Q29. When using the Denman–Beavers iteration for matrix square root, what does it converge to for well-conditioned positive definite A?

• An LU factor of A
• The principal square root of A
• The inverse of A
• A random square root

Correct Answer: The principal square root of A

Q30. If X² = A and X commutes with A (XA = AX), what can be deduced in the diagonalizable case?

• X must be scalar multiple of identity
• X and A can be simultaneously diagonalized, so X is a function of A
• X cannot exist
• X must be nilpotent

Correct Answer: X and A can be simultaneously diagonalized, so X is a function of A

Q31. For Hermitian matrices, how are square roots typically computed?

• Using QR decomposition
• By spectral decomposition: UΛU* and taking Λ^(1/2)
• Using Gaussian elimination elementwise
• By computing the inverse square

Correct Answer: By spectral decomposition: UΛU* and taking Λ^(1/2)

Q32. Which of the following is true about complex matrices and square roots?

• Every complex square matrix has a complex square root
• Some complex matrices do not have any complex square root
• Complex field guarantees a unique square root for all matrices
• Square roots over complex numbers are impossible

Correct Answer: Some complex matrices do not have any complex square root

Q33. If A is positive definite, which square root is usually referred to as the principal square root?

• The one with the largest norm
• The unique positive definite square root
• Any square root with positive determinant
• The zero matrix

Correct Answer: The unique positive definite square root

Q34. Does taking elementwise square root of a matrix A’s entries produce a matrix square root in general?

• Yes, always
• No, elementwise square roots do not generally satisfy X² = A
• Only if A is orthogonal
• Only for triangular matrices

Correct Answer: No, elementwise square roots do not generally satisfy X² = A

Q35. Which condition ensures existence of a unique real symmetric square root for a real symmetric matrix?

• All eigenvalues are nonnegative
• Matrix is singular
• Matrix is skew-symmetric
• Trace is zero

Correct Answer: All eigenvalues are nonnegative

Q36. For application in covariance matrices (statistics in drug studies), why are matrix square roots used?

• To make covariance matrices diagonal
• To compute whitening transformation or factorization for multivariate normal sampling
• They are not used in statistics
• To invert the covariance directly

Correct Answer: To compute whitening transformation or factorization for multivariate normal sampling

Q37. Can a matrix have infinitely many distinct square roots?

• No, at most two
• Yes, some matrices have infinitely many square roots
• Only identity has infinite square roots
• Only zero has infinite square roots

Correct Answer: Yes, some matrices have infinitely many square roots

Q38. Which matrix definitely has a square root equal to itself?

• Any projection matrix P with P² = P
• Any nilpotent matrix
• Any rotation matrix
• Any diagonal matrix with distinct entries

Correct Answer: Any projection matrix P with P² = P

Q39. If A is diagonalizable with eigenvalues λ_i, which choices for eigenvalues μ_i of a square root X are valid?

• μ_i = ±√λ_i for each i, choosing branches independently when allowed
• μ_i = λ_i²
• μ_i = log λ_i
• μ_i = 1/λ_i

Correct Answer: μ_i = ±√λ_i for each i, choosing branches independently when allowed

Q40. Which of these is a practical step before computing a matrix square root numerically?

• Check eigenvalue distribution and conditioning of A
• Replace A with its elementwise absolute values
• Ensure A is upper triangular only
• Force A to be diagonal

Correct Answer: Check eigenvalue distribution and conditioning of A

Q41. For a 3×3 nilpotent matrix N with N³ = 0 but N² ≠ 0, can N have a square root X with X² = N?

• Yes, always
• Not necessarily; existence is constrained by Jordan form and may fail
• No, nilpotent matrices never have square roots
• Only if N is diagonalizable

Correct Answer: Not necessarily; existence is constrained by Jordan form and may fail

Q42. In numerical practice, why might one prefer the principal square root among multiple square roots?

• It is always sparse
• It is Hermitian and positive definite when A is
• It has the largest determinant
• It is not uniquely defined

Correct Answer: It is Hermitian and positive definite when A is

Q43. If A = B² and B is upper triangular, what can be said about A?

• A must be symmetric
• A will also be upper triangular
• A must be diagonal
• A must be singular

Correct Answer: A will also be upper triangular

Q44. Does the existence of a square root for A imply existence of cube roots or higher roots?

• Yes, if square root exists then all higher roots exist
• Not necessarily; existence must be checked for each root order
• Cube roots always exist if square roots do
• No matrix can have roots of different orders

Correct Answer: Not necessarily; existence must be checked for each root order

Q45. For a real positive definite covariance matrix C in pharmacology, which factor can be used for sampling multivariate normals?

• The principal square root C^(1/2)
• The inverse of C only
• Elementwise square roots of C
• Only the identity matrix

Correct Answer: The principal square root C^(1/2)

Q46. If A has an eigenvalue λ = -4, what are possible eigenvalues μ of a complex square root X?

• μ = ±2i
• μ = ±4
• μ = ±√(-4) but not defined
• μ = 0 only

Correct Answer: μ = ±2i

Q47. For numerical stability in computing matrix square roots, which approach is often recommended?

• Direct elementwise operations
• Use stable iterative methods or spectral methods respecting conditioning
• Always use integer arithmetic
• Ignore conditioning and compute directly

Correct Answer: Use stable iterative methods or spectral methods respecting conditioning

Q48. Which of the following is a correct identity if X² = A and X commutes with A?

• X = A
• X = p(A) for some polynomial p (in diagonalizable case)
• X = A⁻¹
• X and A must be diagonal matrices

Correct Answer: X = p(A) for some polynomial p (in diagonalizable case)

Q49. If A is symmetric positive semidefinite, which decomposition is commonly used to find a square root?

• Cholesky decomposition and then take factor as square root
• LU without pivoting
• Elementwise square root
• Compute inverse first

Correct Answer: Cholesky decomposition and then take factor as square root

Q50. Which statement best summarizes practical advice for B. Pharm students dealing with matrix roots?

• Always take elementwise roots of data matrices
• Check matrix type (symmetric, positive definite, diagonalizable), eigenvalues, and conditioning before choosing an analytical or numerical method
• Assume roots are unique and ignore conditioning
• Only computers can find roots; understanding theory is unnecessary

Correct Answer: Check matrix type (symmetric, positive definite, diagonalizable), eigenvalues, and conditioning before choosing an analytical or numerical method

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