Characteristic equation of a square matrix MCQs With Answer

Understanding the characteristic equation of a square matrix is essential for B.Pharm students who study mathematical methods in pharmaceutics and pharmacokinetics. The characteristic equation, obtained from det(A − λI) = 0, links eigenvalues to matrix properties like trace and determinant, aiding analyses of system stability, compartmental models and linear transformations. Mastering how to form and solve characteristic polynomials, interpret algebraic and geometric multiplicities, and apply the Cayley–Hamilton theorem helps in simplifying matrix computations and modeling drug distribution dynamics. This concise, exam-focused guide emphasizes practical steps, common pitfalls, and problem-solving strategies tailored for B.Pharm curricula. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the characteristic equation of a square matrix A?

• det(A − λI) = 0
• det(A + λI) = 0
• trace(A − λI) = 0
• rank(A − λI) = 0

Correct Answer: det(A − λI) = 0

Q2. For an n × n matrix, the characteristic polynomial is of which degree?

• n
• n − 1
• 2n
• 1

Correct Answer: n

Q3. For a 2×2 matrix A = [[a, b], [c, d]], the characteristic equation is:

• λ^2 − (a + d)λ + (ad − bc) = 0
• λ^2 + (a + d)λ + (ad + bc) = 0
• λ^2 − (a − d)λ + (ad − bc) = 0
• λ^2 + (a − d)λ + (ad − bc) = 0

Correct Answer: λ^2 − (a + d)λ + (ad − bc) = 0

Q4. The eigenvalues of a matrix are roots of which expression?

• The characteristic polynomial
• The minimal polynomial
• The adjugate matrix
• The inverse matrix

Correct Answer: The characteristic polynomial

Q5. What is the relationship between the product of eigenvalues and the determinant of A?

• Product of eigenvalues = det(A)
• Product of eigenvalues = trace(A)
• Product of eigenvalues = rank(A)
• Product of eigenvalues = sum of cofactors

Correct Answer: Product of eigenvalues = det(A)

Q6. The sum of eigenvalues of A equals which matrix invariant?

• trace(A)
• det(A)
• rank(A)
• nullity(A)

Correct Answer: trace(A)

Q7. A triangular matrix has characteristic polynomial equal to:

• Product (λ − diagonal entries)
• Product (λ + diagonal entries)
• Sum (λ − diagonal entries)
• Sum (diagonal entries − λ)

Correct Answer: Product (λ − diagonal entries)

Q8. Which theorem states that every square matrix satisfies its own characteristic equation?

• Cayley–Hamilton theorem
• Perron–Frobenius theorem
• Rank–Nullity theorem
• Sylvester’s theorem

Correct Answer: Cayley–Hamilton theorem

Q9. If λ is a root of multiplicity k of the characteristic polynomial, k is called:

• Algebraic multiplicity
• Geometric multiplicity
• Rank multiplicity
• Spectral radius

Correct Answer: Algebraic multiplicity

Q10. Geometric multiplicity of an eigenvalue is:

• Dimension of its eigenspace
• Degree of the minimal polynomial
• Number of Jordan blocks times eigenvalue
• Trace minus determinant

Correct Answer: Dimension of its eigenspace

Q11. If algebraic multiplicity > geometric multiplicity for an eigenvalue, the matrix is:

• Defective (not diagonalizable)
• Diagonalizable
• Orthogonal
• Singular

Correct Answer: Defective (not diagonalizable)

Q12. The characteristic polynomial of I_n (identity matrix) is:

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

Correct Answer: (1 − λ)^n

Q13. For scalar c and matrix A, characteristic polynomial of cA relates how to that of A?

• P_cA(λ) = c^n P_A(λ/c)
• P_cA(λ) = P_A(λ − c)
• P_cA(λ) = P_A(λ)/c
• P_cA(λ) = P_A(cλ)

Correct Answer: P_cA(λ) = c^n P_A(λ/c)

Q14. Similar matrices have characteristic polynomials that are:

• Identical
• Reciprocals
• Transposes
• Negatives

Correct Answer: Identical

Q15. The characteristic polynomial is always a:

• Monic polynomial of degree n
• Polynomial with constant leading coefficient
• Polynomial of degree n−1
• Non-monic polynomial of degree n

Correct Answer: Monic polynomial of degree n

Q16. If 0 is an eigenvalue of A, what does it imply about A?

• A is singular (non-invertible)
• A is orthogonal
• A is diagonalizable
• A has full rank

Correct Answer: A is singular (non-invertible)

Q17. For a nilpotent matrix N (N^k = 0 for some k), its only eigenvalue is:

• 0
• 1
• −1
• Depends on k

Correct Answer: 0

Q18. The minimal polynomial divides the characteristic polynomial and shares the same:

• Eigenvalues
• Algebraic multiplicities
• Degree
• Leading coefficient

Correct Answer: Eigenvalues

Q19. Characteristic polynomial of A^T (transpose) equals:

• Characteristic polynomial of A
• Negative of characteristic polynomial of A
• Reciprocal of characteristic polynomial of A
• Square of characteristic polynomial of A

Correct Answer: Characteristic polynomial of A

Q20. If A is block diagonal with blocks B and C, characteristic polynomial of A is:

• Product of characteristic polynomials of B and C
• Sum of characteristic polynomials of B and C
• Characteristic polynomial of B minus that of C
• Characteristic polynomial of B transposed with C

Correct Answer: Product of characteristic polynomials of B and C

Q21. Which method directly computes the characteristic polynomial?

• Compute det(A − λI)
• Compute inverse of A
• Compute A^2
• Compute trace(A^2)

Correct Answer: Compute det(A − λI)

Q22. For a 3×3 matrix, the coefficient of λ^2 in characteristic polynomial equals:

• −trace(A)
• trace(A)
• det(A)
• Sum of principal minors

Correct Answer: −trace(A)

Q23. In pharmacokinetic compartment models represented by matrices, eigenvalues help determine:

• Rates of exponential decay or growth
• Molecular weight of drug
• Solubility of compound
• Viscosity of medium

Correct Answer: Rates of exponential decay or growth

Q24. For a 2×2 rotation matrix, eigenvalues are complex; what does characteristic equation reveal?

• Complex conjugate roots on unit circle
• Real repeated roots
• Zero eigenvalue
• Infinite eigenvalues

Correct Answer: Complex conjugate roots on unit circle

Q25. If eigenvalues of A are λ1, λ2, …, λn, the characteristic polynomial equals:

• ∏(λi − λ) with sign convention
• ∑(λi + λ)
• ∏(λi + λ)
• ∑(λi − λ)

Correct Answer: ∏(λi − λ) with sign convention

Q26. Which property is NOT determined solely by the characteristic polynomial?

• Geometric multiplicity of eigenvalues
• Eigenvalues (with algebraic multiplicity)
• Determinant
• Trace

Correct Answer: Geometric multiplicity of eigenvalues

Q27. For matrix A, characteristic polynomial evaluated at 0 equals:

• (−1)^n det(A)
• det(A)
• trace(A)
• 0 always

Correct Answer: (−1)^n det(A)

Q28. In clinical modeling, repeated eigenvalues can indicate:

• Modes with repeated time-constants
• Immediate elimination only
• No steady state
• Negative concentrations

Correct Answer: Modes with repeated time-constants

Q29. For a diagonalizable matrix, algebraic and geometric multiplicities are:

• Equal for each eigenvalue
• Always different
• Algebraic multiplicity is zero
• Geometric multiplicity always greater

Correct Answer: Equal for each eigenvalue

Q30. The characteristic polynomial of a 3×3 matrix A is λ^3 − c2 λ^2 + c1 λ − c0. What is c0?

• det(A)
• trace(A)
• sum of eigenvalues
• rank(A)

Correct Answer: det(A)

Q31. If A has characteristic polynomial (λ − 2)^3, then 2 is an eigenvalue with algebraic multiplicity:

• 3
• 1
• 2
• 0

Correct Answer: 3

Q32. Which matrix operation does NOT change the characteristic polynomial?

• Replacing A by P^−1AP (similarity)
• Adding a scalar to one row
• Multiplying a row by 2
• Replacing A by its transpose

Correct Answer: Replacing A by P^−1AP (similarity)

Q33. For companion matrix of monic polynomial, characteristic polynomial equals:

• Given monic polynomial
• Reciprocal polynomial
• Zero polynomial
• Polynomial squared

Correct Answer: Given monic polynomial

Q34. The spectral radius of A is defined as:

• Maximum absolute value of eigenvalues
• Sum of eigenvalues
• Minimum eigenvalue
• Determinant divided by trace

Correct Answer: Maximum absolute value of eigenvalues

Q35. Which statement about characteristic polynomial coefficients is true?

• They are (up to sign) elementary symmetric functions of eigenvalues
• They equal eigenvectors
• They always vanish for singular matrices
• They are independent of A

Correct Answer: They are (up to sign) elementary symmetric functions of eigenvalues

Q36. For continuous-time linear systems x’ = Ax, eigenvalues from characteristic equation determine:

• Stability of equilibrium
• Mass balance of drug
• Solubility limit
• Boiling point

Correct Answer: Stability of equilibrium

Q37. If A has characteristic polynomial λ^2 + 1, eigenvalues are:

• ±i
• ±1
• 0, 0
• 1, −1

Correct Answer: ±i

Q38. Which matrix has characteristic polynomial λ^n?

• Zero matrix of size n
• Identity matrix
• Diagonal with ones
• Any invertible matrix

Correct Answer: Zero matrix of size n

Q39. For a 3×3 triangular matrix, characteristic roots are:

• Its diagonal entries
• Its row sums
• Its column sums
• Zeros only

Correct Answer: Its diagonal entries

Q40. The characteristic polynomial is invariant under which change?

• Similarity transformation
• Arbitrary row addition
• Multiplying a row by nonzero scalar
• Replacing A with A + I

Correct Answer: Similarity transformation

Q41. If A is singular, characteristic polynomial evaluated at zero equals:

• 0
• 1
• Trace(A)
• −1

Correct Answer: 0

Q42. How does one check if λ0 is an eigenvalue using characteristic equation?

• Substitute λ0 into det(A − λI) and check if zero
• Compute A^−1 and check entry (λ0)
• Compute trace(A) − λ0
• Compute determinant of A + λ0 I

Correct Answer: Substitute λ0 into det(A − λI) and check if zero

Q43. If A has characteristic polynomial p(λ), Cayley–Hamilton implies p(A) equals:

• Zero matrix
• Identity matrix
• Determinant times identity
• A^−1

Correct Answer: Zero matrix

Q44. Which matrix property can be deduced directly from characteristic polynomial coefficients?

• Trace and determinant
• Geometric multiplicity
• Number of Jordan blocks
• Eigenvectors

Correct Answer: Trace and determinant

Q45. For real matrices, complex eigenvalues appear in:

• Conjugate pairs
• Triplets
• Singletons only
• Quads only

Correct Answer: Conjugate pairs

Q46. If A is 2×2 with trace t and determinant d, characteristic equation is:

• λ^2 − tλ + d = 0
• λ^2 + tλ + d = 0
• λ^2 − dλ + t = 0
• λ^2 + dλ + t = 0

Correct Answer: λ^2 − tλ + d = 0

Q47. For matrix A, multiplicity of eigenvalue in characteristic polynomial is algebraic multiplicity; geometric multiplicity ≤ algebraic multiplicity. True or false?

• True
• False
• Only for symmetric matrices
• Only for diagonal matrices

Correct Answer: True

Q48. In pharmacokinetic linear system matrices, diagonalization simplifies:

• Solving system of ODEs by decoupling modes
• Measuring pH
• Determining solubility
• Calculating molecular structure

Correct Answer: Solving system of ODEs by decoupling modes

Q49. Which of the following alters the characteristic polynomial by shifting λ to λ − c?

• Replacing A by A + cI
• Multiplying A by c
• Transposing A
• Taking inverse of A

Correct Answer: Replacing A by A + cI

Q50. For a real symmetric matrix, characteristic polynomial roots are:

• Real
• Complex with nonzero imaginary part
• Always positive
• Always negative

Correct Answer: Real

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