Cayley–Hamilton theorem MCQs With Answer

Introduction: Mastering the Cayley–Hamilton theorem is essential for B. Pharm students studying linear algebra applications in pharmacokinetics, compartmental modeling, and drug distribution analysis. This collection of Cayley–Hamilton theorem MCQs With Answer focuses on core ideas—characteristic polynomial, matrix powers, minimal polynomial, and matrix exponential—using pharmacy-relevant examples and clear, exam-oriented practice. Questions go deeper than definitions, covering verification of the theorem on small matrices, techniques to compute A^n, implications for diagonalizable and Jordan-block matrices, and applications to linear ODE systems used in compartment models. Strengthen your analytical skills and problem-solving confidence for coursework and exams. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the Cayley–Hamilton theorem?

• Every matrix satisfies its own characteristic polynomial
• Every vector is an eigenvector of some matrix
• Every matrix is diagonalizable
• Every polynomial has a matrix root

Correct Answer: Every matrix satisfies its own characteristic polynomial

Q2. For a 2×2 matrix A with characteristic polynomial λ^2 − (tr A)λ + det A, what does Cayley–Hamilton give for A?

• A^2 − (tr A)A + (det A)I = 0
• A^2 + (tr A)A + (det A)I = 0
• A^2 − (det A)A + (tr A)I = 0
• A^2 = (tr A)A − (det A)I

Correct Answer: A^2 − (tr A)A + (det A)I = 0

Q3. If A is a diagonal matrix with diagonal entries 2 and 3, what is its characteristic polynomial?

• (λ − 2)(λ − 3)
• (λ + 2)(λ + 3)
• (λ − 5)
• λ^2 − 6λ + 5 cannot be characteristic

Correct Answer: (λ − 2)(λ − 3)

Q4. Using Cayley–Hamilton, which expression can be used to compute A^n for a 2×2 matrix?

• Express A^n as a linear combination of I and A
• Express A^n as a polynomial of degree n in A without reduction
• Only diagonalization can compute A^n
• A^n must be computed by repeated multiplication

Correct Answer: Express A^n as a linear combination of I and A

Q5. For a 3×3 matrix, Cayley–Hamilton lets you reduce powers of A to

• A quadratic polynomial in A at most
• A cubic polynomial only
• A constant times I only
• A polynomial of degree 4

Correct Answer: A quadratic polynomial in A at most

Q6. Which polynomial is substituted into the matrix in Cayley–Hamilton theorem?

• The characteristic polynomial of the matrix
• The minimal polynomial of the matrix
• Any arbitrary polynomial
• The resolvent polynomial

Correct Answer: The characteristic polynomial of the matrix

Q7. If the characteristic polynomial of A is p(λ) = λ^2 − 5λ + 6, what is p(A)?

• A^2 − 5A + 6I
• A^2 + 5A + 6I
• A^2 − 6A + 5I
• A^2 − 5A − 6I

Correct Answer: A^2 − 5A + 6I

Q8. The minimal polynomial of a matrix divides the characteristic polynomial. What is a consequence for Cayley–Hamilton?

• The minimal polynomial also annihilates A
• The characteristic polynomial never annihilates A
• A is always diagonalizable
• A has distinct eigenvalues

Correct Answer: The minimal polynomial also annihilates A

Q9. Which of these is a typical application of Cayley–Hamilton in pharmacy math?

• Computing state transition matrices for compartmental models
• Designing pill coatings
• Measuring pH of solutions
• Calculating molecular weight

Correct Answer: Computing state transition matrices for compartmental models

Q10. If A has characteristic polynomial λ^2 − 4λ + 3, what is A^2 in terms of A and I?

• A^2 = 4A − 3I
• A^2 = 3A − 4I
• A^2 = 4I − 3A
• A^2 = A + 3I

Correct Answer: A^2 = 4A − 3I

Q11. If eigenvalues of A are 1 and 2, what is tr A and det A?

• tr A = 3 and det A = 2
• tr A = 2 and det A = 3
• tr A = 1 and det A = 2
• tr A = 3 and det A = 1

Correct Answer: tr A = 3 and det A = 2

Q12. For a 2×2 matrix A, Cayley–Hamilton implies A^3 can be written as

• αA + βI for some scalars α, β
• A^3 is independent and cannot be simplified
• αA^2 + βA + γI with γ ≠ 0 always
• A^3 = 0 for nonzero A

Correct Answer: αA + βI for some scalars α, β

Q13. True or false: Cayley–Hamilton holds for all square matrices over any field.

• True
• False
• True only over R
• True only for diagonalizable matrices

Correct Answer: True

Q14. How does Cayley–Hamilton help compute e^{At} for linear ODE systems?

• It expresses powers of A in a finite basis, enabling polynomial approximation of e^{At}
• It diagonalizes A automatically
• It computes e^{At} by series without reduction
• It gives eigenvectors directly

Correct Answer: It expresses powers of A in a finite basis, enabling polynomial approximation of e^{At}

Q15. If A is 2×2 with characteristic polynomial λ^2 − 2λ + 5, is A diagonalizable over R?

• No, because eigenvalues are complex
• Yes, because characteristic polynomial has degree 2
• Yes, because trace is nonzero
• No, because determinant is positive

Correct Answer: No, because eigenvalues are complex

Q16. For matrix A, if p(λ) is the characteristic polynomial, what is p(0)?

• ±det A depending on sign convention
• tr A
• Always zero
• Inverse of det A

Correct Answer: ±det A depending on sign convention

Q17. Which matrix must always satisfy its characteristic polynomial according to Cayley–Hamilton?

• Any square matrix A
• Only symmetric matrices
• Only invertible matrices
• Only diagonal matrices

Correct Answer: Any square matrix A

Q18. If the characteristic polynomial is (λ − 2)^2, which of the following may be true?

• The minimal polynomial is (λ − 2) or (λ − 2)^2
• The minimal polynomial must be (λ − 2)^3
• The matrix is necessarily diagonalizable
• The eigenvalues are 2 and 3

Correct Answer: The minimal polynomial is (λ − 2) or (λ − 2)^2

Q19. How can Cayley–Hamilton be used to invert an invertible matrix A?

• Use characteristic polynomial to express I as polynomial in A then solve for A^{-1}
• It cannot be used to find inverses
• It directly gives eigenvectors which are inverses
• It gives determinant only

Correct Answer: Use characteristic polynomial to express I as polynomial in A then solve for A^{-1}

Q20. If A satisfies A^2 = 3A − 2I, what is the characteristic polynomial of A (up to scalar)?

• λ^2 − 3λ + 2
• λ^2 + 3λ − 2
• λ^2 − 2λ + 3
• λ^2 + 2λ + 3

Correct Answer: λ^2 − 3λ + 2

Q21. For companion matrix C of polynomial λ^2 − aλ − b, Cayley–Hamilton ensures p(C)=0. What is the main use of companion matrices?

• To represent polynomials as matrices and study roots
• To compute determinants faster
• To diagonalize any matrix
• To compute integrals in ODEs

Correct Answer: To represent polynomials as matrices and study roots

Q22. Which statement is true regarding the degree of the minimal polynomial m(λ) of an n×n matrix?

• deg m(λ) ≤ n
• deg m(λ) ≥ n
• deg m(λ) = n always
• deg m(λ) = 1 always

Correct Answer: deg m(λ) ≤ n

Q23. For A = [[0,1],[-2,3]], what is the characteristic polynomial?

• λ^2 − 3λ + 2
• λ^2 + 3λ + 2
• λ^2 − 2λ + 3
• λ^2 + 2λ − 3

Correct Answer: λ^2 − 3λ + 2

Q24. Using Cayley–Hamilton, which basis spans all powers of a 3×3 matrix A?

• I, A, A^2
• I, A only
• A^2, A^3, A^4
• Only I

Correct Answer: I, A, A^2

Q25. In pharmacokinetic compartment models, why is computing e^{At} important?

• It gives the state transition matrix for concentration evolution
• It gives the pKa of a drug
• It measures bioavailability directly
• It computes molecular interactions

Correct Answer: It gives the state transition matrix for concentration evolution

Q26. If A has characteristic polynomial λ^3 − c2λ^2 + c1λ − c0, then Cayley–Hamilton implies which relation?

• A^3 − c2A^2 + c1A − c0I = 0
• A^3 + c2A^2 + c1A + c0I = 0
• A^3 − c0A^2 + c1A − c2I = 0
• A^2 − c2A + c1I = 0

Correct Answer: A^3 − c2A^2 + c1A − c0I = 0

Q27. If A is nilpotent (A^k = 0 for some k), what does Cayley–Hamilton imply about its characteristic polynomial?

• Constant term is 0, so det A = 0
• All coefficients are zero
• Characteristic polynomial is 1
• Matrix must be identity

Correct Answer: Constant term is 0, so det A = 0

Q28. For a 2×2 matrix with trace t and determinant d, characteristic polynomial is λ^2 − tλ + d. Which scalar relation follows from Cayley–Hamilton?

• A^2 = tA − dI
• A^2 = dA − tI
• A^2 = tI − dA
• A^2 = A + I

Correct Answer: A^2 = tA − dI

Q29. If eigenvalues of A are λ1, λ2, λ3 (not necessarily distinct), what is det A?

• λ1λ2λ3
• λ1 + λ2 + λ3
• λ1^2 + λ2^2 + λ3^2
• λ1/λ2/λ3

Correct Answer: λ1λ2λ3

Q30. Which method complements Cayley–Hamilton to compute functions f(A) like sin(A) or e^{A}?

• Use polynomial reduction combined with Taylor series coefficients
• Use LU decomposition only
• Use row-reduction to RREF only
• Use direct element-wise function application

Correct Answer: Use polynomial reduction combined with Taylor series coefficients

Q31. If A has characteristic polynomial (λ − 1)(λ − 2)^2, what is the degree of its minimal polynomial?

• 2 or 3 depending on Jordan blocks
• Always 1
• Always 2
• Always 3

Correct Answer: 2 or 3 depending on Jordan blocks

Q32. For matrix A = [[1,0],[0,1]], what is its characteristic polynomial?

• (λ − 1)^2
• (λ + 1)^2
• λ^2 − 1
• λ^2

Correct Answer: (λ − 1)^2

Q33. Which of the following is a direct computational benefit of Cayley–Hamilton?

• Reducing computing A^n to computing a fixed small set of matrix powers
• Direct computation of eigenvectors without solving linear equations
• Faster determinant by inspection
• Automatic diagonalization of A

Correct Answer: Reducing computing A^n to computing a fixed small set of matrix powers

Q34. If A is 3×3 and satisfies A^3 = I, what can be said about its characteristic polynomial?

• Its eigenvalues are cube roots of unity
• Its eigenvalues are all 1 only
• It must be diagonalizable with eigenvalue 0
• Its determinant is zero

Correct Answer: Its eigenvalues are cube roots of unity

Q35. In verifying Cayley–Hamilton for a 2×2 matrix numerically, which step is essential?

• Compute characteristic polynomial, substitute A and verify zero matrix
• Only compute eigenvalues numerically
• Compute inverse of A first
• Compute singular values only

Correct Answer: Compute characteristic polynomial, substitute A and verify zero matrix

Q36. If A has characteristic polynomial λ^2 − 6λ + 9, which of the following is true?

• The polynomial factors as (λ − 3)^2, so eigenvalue 3 has algebraic multiplicity 2
• Eigenvalues are 6 and 9
• Matrix is singular with determinant 0
• Trace is 9 and determinant is 6

Correct Answer: The polynomial factors as (λ − 3)^2, so eigenvalue 3 has algebraic multiplicity 2

Q37. Which relation helps reduce A^4 for a 3×3 matrix using Cayley–Hamilton?

• Use A^3 expressed from characteristic polynomial, then multiply by A
• Compute A^4 directly by multiplication only
• Use A^4 = I always
• A^4 cannot be reduced

Correct Answer: Use A^3 expressed from characteristic polynomial, then multiply by A

Q38. For a matrix used in a two-compartment model, why might Cayley–Hamilton be practically useful?

• It simplifies repeated powers of the transfer matrix when computing time evolution
• It measures drug potency directly
• It changes compartments’ physical properties
• It replaces biochemical assays

Correct Answer: It simplifies repeated powers of the transfer matrix when computing time evolution

Q39. If p(λ) = λ^2 + 1 is characteristic polynomial, what does p(A)=0 imply?

• A^2 = −I
• A^2 = I
• A is zero matrix
• A has eigenvalues 0 and ±i only

Correct Answer: A^2 = −I

Q40. For a 2×2 rotation matrix R(θ), its characteristic polynomial is λ^2 − 2 cosθ λ + 1. What does Cayley–Hamilton give?

• R^2 − 2 cosθ R + I = 0
• R^2 + 2 cosθ R + I = 0
• R^2 = I only
• R has no characteristic polynomial

Correct Answer: R^2 − 2 cosθ R + I = 0

Q41. When applying Cayley–Hamilton to non-diagonalizable A, what is required?

• Nothing special; theorem still holds
• Diagonalization first
• A must be symmetric
• A must be normal

Correct Answer: Nothing special; theorem still holds

Q42. How does the companion matrix construction connect to Cayley–Hamilton?

• Companion matrix is a matrix whose characteristic polynomial equals the given polynomial
• Companion matrix diagonalizes any polynomial
• It computes integrals in pharmacology directly
• It gives the minimal polynomial only

Correct Answer: Companion matrix is a matrix whose characteristic polynomial equals the given polynomial

Q43. If A is invertible and p(λ) is the characteristic polynomial, which relationship helps find A^{-1}?

• Divide p(A)=0 by A then rearrange to express A^{-1} as polynomial in A
• p(λ) gives eigenvectors which are inverses
• A^{-1} = p(A) always
• There is no relation to A^{-1}

Correct Answer: Divide p(A)=0 by A then rearrange to express A^{-1} as polynomial in A

Q44. If A has characteristic polynomial λ^2 − λ, what is true about A?

• A(A − I) = 0, so eigenvalues 0 and/or 1
• A is invertible with inverse I
• Eigenvalues are ±1 only
• A has determinant 1 only

Correct Answer: A(A − I) = 0, so eigenvalues 0 and/or 1

Q45. Which computational step is NOT needed to apply Cayley–Hamilton to a given small matrix?

• Finding characteristic polynomial
• Substituting A into its polynomial
• Computing eigenvectors explicitly
• Computing powers of A up to degree n−1

Correct Answer: Computing eigenvectors explicitly

Q46. For A = [[0,1],[−1,0]], what does Cayley–Hamilton say about A^2?

• A^2 = −I
• A^2 = I
• A^2 = 0
• A^2 = A

Correct Answer: A^2 = −I

Q47. In practice, which numerical issue can affect using Cayley–Hamilton for large matrices?

• Round-off error in computing characteristic polynomial and coefficients
• Cayley–Hamilton is invalid numerically
• It gives wrong eigenvalues always
• It causes overflow of matrix size

Correct Answer: Round-off error in computing characteristic polynomial and coefficients

Q48. If characteristic polynomial of A is λ^3, what does Cayley–Hamilton imply?

• A^3 = 0, so A is nilpotent of index ≤ 3
• A is identity
• A has full rank
• A has an inverse

Correct Answer: A^3 = 0, so A is nilpotent of index ≤ 3

Q49. For a pharmacokinetic teacher, which explanation best conveys Cayley–Hamilton to students?

• Every square transition matrix satisfies its characteristic polynomial, letting you reduce high matrix powers in time-evolution calculations
• It is a rule to compute drug concentration directly from polynomial formulas without matrices
• It only applies to scalar ODEs
• It replaces all need for numerical simulation

Correct Answer: Every square transition matrix satisfies its characteristic polynomial, letting you reduce high matrix powers in time-evolution calculations

Q50. Which is a concise summary of Cayley–Hamilton useful for exam answers?

• A satisfies p_A(A) = 0 where p_A is characteristic polynomial of A
• A is always equal to its characteristic polynomial
• Characteristic polynomial equals trace times determinant
• Every polynomial annihilates every matrix

Correct Answer: A satisfies p_A(A) = 0 where p_A is characteristic polynomial of A

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