Matrices - characteristic roots, Cayley–Hamilton theorem, canonical forms under equivalence - Question Bank
1. The characteristic roots of A and A⁻¹ (if A is invertible) are:
2. The characteristic roots of A and A^T are:
3. The set of all matrices similar to a given matrix A forms a:
4. The set of all matrices equivalent to a given matrix A forms a:
5. If a matrix A is similar to a diagonal matrix D, then A is called:
6. The canonical form of a matrix under equivalence is determined by its:
7. According to the Cayley-Hamilton theorem, for A = [[2, 1], [1, 2]], we have:
8. The eigenvalues of A = [[2, 1], [1, 2]] are:
9. If A = [[2, 1], [1, 2]], its characteristic equation is:
10. The Cayley-Hamilton theorem can be used to:
11. If A is a singular matrix, then at least one of its eigenvalues is:
12. The characteristic equation of a 2x2 matrix [[a, b], [c, d]] is:
13. If A and B are equivalent matrices, they have the same:
14. The canonical form under congruence transformation for a real symmetric matrix is:
15. The Jordan Normal Form is a canonical form under which transformation?
16. A matrix A is diagonalizable if and only if:
17. If A is a nilpotent matrix, then all its eigenvalues are:
18. 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:
19. Which of the following is NOT preserved under equivalence transformations?
20. Which of the following is NOT preserved under similarity transformations?
21. The canonical form under similarity is obtained by applying:
22. The canonical form under equivalence is obtained by applying:
23. If A is an orthogonal matrix, its eigenvalues have absolute value:
24. If A is a skew-symmetric matrix with real entries, its eigenvalues are:
25. If A is a real symmetric matrix, its eigenvalues are:
26. The set of all eigenvalues of a matrix A is called its:
27. 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:
28. According to the Cayley-Hamilton theorem, the characteristic polynomial p(λ) = det(A - λI) satisfies:
29. What is the minimum polynomial of a matrix A?
30. If A is a diagonal matrix, its eigenvalues are:
31. The characteristic roots of a matrix are also known as:
32. Two matrices A and B are similar if there exists an invertible matrix P such that:
33. If two matrices are similar, they have the same:
34. The canonical form under similarity transformation is:
35. The invariant factors of a matrix are:
36. Two matrices A and B are equivalent if there exists an invertible matrix P such that:
37. What is the Smith Normal Form of a matrix?
38. The canonical form of a matrix under equivalence transformation is unique if the matrix is:
39. If A is a 2x2 matrix with eigenvalues λ1 and λ2, then det(A) is equal to:
40. If A is a 2x2 matrix with eigenvalues λ1 and λ2, then tr(A) is equal to:
41. Which theorem guarantees that a matrix is a root of its characteristic polynomial?
42. For an n x n matrix A, how many eigenvalues can it have?
43. The scalar λ in the equation Av = λv is called the:
44. What is an eigenvector of a matrix A?
45. If A is an n x n matrix, its characteristic polynomial has a degree of:
46. What is the determinant of a matrix A if λ = 0 is one of its eigenvalues?
47. What is the trace of a square matrix?
48. If A is a square matrix, what does the Cayley-Hamilton theorem state about A?
49. According to the Cayley-Hamilton theorem, every square matrix satisfies its own:
50. The roots of the characteristic equation of a matrix are called: