TRB PG Assistant - Matrices - characteristic roots, Cayley–Hamilton theorem, canonical forms under equivalence Online Test (MCQ Quiz)

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

A) det(A - λI) = 0 B) det(A + λI) = 0 C) det(λA - I) = 0 D) det(λI - A) = 0

2. The roots of the characteristic equation of a matrix are called:

A) Eigenvalues B) Singular values C) Roots of unity D) Determinants

3. According to the Cayley-Hamilton theorem, every square matrix satisfies its own:

A) Characteristic equation B) Minimal polynomial C) Eigenvector equation D) Eigenvalue equation

4. If A is a square matrix, what does the Cayley-Hamilton theorem state about A?

A) A satisfies its characteristic polynomial, i.e., p(A) = 0 B) A is always diagonalizable C) The determinant of A is always zero D) The trace of A is always zero

5. What is the trace of a square matrix?

A) The sum of its diagonal elements B) The product of its diagonal elements C) The determinant of the matrix D) The sum of all its elements

6. What is the determinant of a matrix A if λ = 0 is one of its eigenvalues?

A) 0 B) 1 C) The product of non-zero eigenvalues D) The sum of eigenvalues

7. If A is an n x n matrix, its characteristic polynomial has a degree of:

A) n B) n-1 C) n+1 D) 1

8. What is an eigenvector of a matrix A?

A) A non-zero vector v such that Av = λv for some scalar λ B) A vector v such that Av = 0 C) A vector v such that v^T A = 0 D) A vector v such that Av = v

9. The scalar λ in the equation Av = λv is called the:

A) Eigenvalue B) Eigenvector C) Characteristic root D) Scalar multiplier

10. For an n x n matrix A, how many eigenvalues can it have?

A) At most n B) Exactly n C) Exactly n^2 D) At least n

11. Which theorem guarantees that a matrix is a root of its characteristic polynomial?

A) Cayley-Hamilton theorem B) Schur's theorem C) Perron-Frobenius theorem D) Spectral theorem

12. If A is a 2x2 matrix with eigenvalues λ1 and λ2, then tr(A) is equal to:

A) λ1 + λ2 B) λ1 * λ2 C) λ1 - λ2 D) (λ1 + λ2) / 2

13. If A is a 2x2 matrix with eigenvalues λ1 and λ2, then det(A) is equal to:

A) λ1 * λ2 B) λ1 + λ2 C) λ1 - λ2 D) (λ1 + λ2) / 2

14. The canonical form of a matrix under equivalence transformation is unique if the matrix is:

A) In Smith Normal Form B) Diagonalizable C) Symmetric D) Hermitian

15. What is the Smith Normal Form of a matrix?

A) A diagonal matrix with invariant factors on the diagonal B) A matrix with all ones on the diagonal C) A matrix similar to the original matrix D) A matrix derived from row and column operations

16. Two matrices A and B are equivalent if there exists an invertible matrix P such that:

A) B = P⁻¹AP B) B = PAP⁻¹ C) B = P⁻¹PA D) B = AP⁻¹P

17. The invariant factors of a matrix are:

A) Unique polynomials that are preserved under elementary row and column operations B) The eigenvalues of the matrix C) The eigenvectors of the matrix D) The entries of the matrix

18. The canonical form under similarity transformation is:

A) Jordan Normal Form B) Smith Normal Form C) Row Echelon Form D) Reduced Row Echelon Form

19. If two matrices are similar, they have the same:

A) Eigenvalues B) Determinant C) Trace D) All of the above

20. Two matrices A and B are similar if there exists an invertible matrix P such that:

A) B = P⁻¹AP B) B = PAP⁻¹ C) B = P⁻¹PA D) B = AP⁻¹P

21. The characteristic roots of a matrix are also known as:

A) Eigenvalues B) Latent roots C) Principal values D) All of the above

22. If A is a diagonal matrix, its eigenvalues are:

A) The diagonal entries B) The off-diagonal entries C) Zero D) One

23. What is the minimum polynomial of a matrix A?

A) The monic polynomial of least degree that annihilates A B) The characteristic polynomial of A C) The polynomial whose roots are the eigenvalues of A D) The polynomial of degree 1 that annihilates A

24. According to the Cayley-Hamilton theorem, the characteristic polynomial p(λ) = det(A - λI) satisfies:

A) p(A) = 0 B) p(A) = I C) p(A) = A D) p(A) = det(A)

25. If A is an n x n matrix, and λ1, λ2, ..., λn are its eigenvalues (counting multiplicity), then tr(A) = Σ(λi) and det(A) = Π(λi). This is a consequence of:

A) The properties of characteristic roots B) The Cayley-Hamilton theorem C) The definition of equivalence D) The definition of similarity

26. The set of all eigenvalues of a matrix A is called its:

A) Spectrum B) Characteristic set C) Eigen-set D) Root set

27. If A is a real symmetric matrix, its eigenvalues are:

A) Real B) Complex C) Imaginary D) Zero

28. If A is a skew-symmetric matrix with real entries, its eigenvalues are:

A) Purely imaginary or zero B) Real C) Complex D) Positive

29. If A is an orthogonal matrix, its eigenvalues have absolute value:

A) 1 B) 0 C) Greater than 1 D) Less than 1

30. The canonical form under equivalence is obtained by applying:

A) Elementary row and column operations B) Similarity transformations C) Orthogonal transformations D) Unitary transformations

31. The canonical form under similarity is obtained by applying:

A) Similarity transformations B) Elementary row and column operations C) Orthogonal transformations D) Unitary transformations

32. Which of the following is NOT preserved under similarity transformations?

A) Eigenvalues B) Determinant C) Trace D) Rank

33. Which of the following is NOT preserved under equivalence transformations?

A) Rank B) Determinant C) Trace D) Eigenvalues

34. The characteristic polynomial of a 3x3 matrix A is given by -λ³ + tr(A)λ² - (sum of principal minors of order 2)λ + det(A). This is consistent with:

A) The properties of eigenvalues and the characteristic equation B) The Cayley-Hamilton theorem C) Smith Normal Form D) Jordan Normal Form

35. If A is a nilpotent matrix, then all its eigenvalues are:

A) Zero B) One C) Real D) Imaginary

36. A matrix A is diagonalizable if and only if:

A) Its minimal polynomial has distinct roots B) Its characteristic polynomial has distinct roots C) Its determinant is non-zero D) Its trace is non-zero

37. The Jordan Normal Form is a canonical form under which transformation?

A) Similarity B) Equivalence C) Congruence D) Unitary equivalence

38. The canonical form under congruence transformation for a real symmetric matrix is:

A) A diagonal matrix with +1, -1, and 0 on the diagonal B) A diagonal matrix with eigenvalues on the diagonal C) Jordan Normal Form D) Smith Normal Form

39. If A and B are equivalent matrices, they have the same:

A) Rank B) Determinant C) Trace D) Eigenvalues

40. The characteristic equation of a 2x2 matrix [[a, b], [c, d]] is:

A) λ² - (a+d)λ + (ad-bc) = 0 B) λ² + (a+d)λ + (ad-bc) = 0 C) λ² - (a+d)λ - (ad-bc) = 0 D) λ² + (a+d)λ - (ad-bc) = 0

41. If A is a singular matrix, then at least one of its eigenvalues is:

A) 0 B) 1 C) Non-zero D) Complex

42. The Cayley-Hamilton theorem can be used to:

A) Find the inverse of a matrix B) Find the determinant of a matrix C) Find the eigenvalues of a matrix D) All of the above

43. If A = [[2, 1], [1, 2]], its characteristic equation is:

A) λ² - 4λ + 3 = 0 B) λ² + 4λ + 3 = 0 C) λ² - 3λ + 4 = 0 D) λ² + 3λ + 4 = 0

44. The eigenvalues of A = [[2, 1], [1, 2]] are:

A) 1 and 3 B) 2 and 2 C) -1 and 3 D) 1 and 1

45. According to the Cayley-Hamilton theorem, for A = [[2, 1], [1, 2]], we have:

A) A² - 4A + 3I = 0 B) A² + 4A + 3I = 0 C) A² - 3A + 4I = 0 D) A² + 3A + 4I = 0

46. The canonical form of a matrix under equivalence is determined by its:

A) Invariant factors B) Eigenvalues C) Minimal polynomial D) Characteristic polynomial

47. If a matrix A is similar to a diagonal matrix D, then A is called:

A) Diagonalizable B) Equipotent C) Orthogonal D) Normal

48. The set of all matrices equivalent to a given matrix A forms a:

A) Class B) Group C) Ring D) Field

49. The set of all matrices similar to a given matrix A forms a:

A) Class B) Group C) Ring D) Field

50. The characteristic roots of A and A^T are:

A) The same B) Opposite C) Reciprocal D) Unrelated

Matrices - characteristic roots, Cayley–Hamilton theorem, canonical forms under equivalence - Online Test

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