Matrices - characteristic roots, Cayley–Hamilton theorem, canonical forms under equivalence - One Line Questions

1. If A is a singular matrix, then at least one of its eigenvalues is: — 0
2. What is the determinant of a matrix A if λ = 0 is one of its eigenvalues? — 0
3. If A is an orthogonal matrix, its eigenvalues have absolute value: — 1
4. The eigenvalues of A = [[2, 1], [1, 2]] are: — 1 and 3
5. The canonical form under congruence transformation for a real symmetric matrix is: — A diagonal matrix with +1, -1, and 0 on the diagonal
6. What is the Smith Normal Form of a matrix? — A diagonal matrix with invariant factors on the diagonal
7. What is an eigenvector of a matrix A? — A non-zero vector v such that Av = λv for some scalar λ
8. If A is a square matrix, what does the Cayley-Hamilton theorem state about A? — A satisfies its characteristic polynomial, i.e., p(A) = 0
9. According to the Cayley-Hamilton theorem, for A = [[2, 1], [1, 2]], we have: — A² - 4A + 3I = 0
10. For an n x n matrix A, how many eigenvalues can it have? — Exactly n
11. Two matrices A and B are equivalent if there exists an invertible matrix P such that: — B = PAP⁻¹
12. Two matrices A and B are similar if there exists an invertible matrix P such that: — B = P⁻¹AP
13. Which theorem guarantees that a matrix is a root of its characteristic polynomial? — Cayley-Hamilton theorem
14. According to the Cayley-Hamilton theorem, every square matrix satisfies its own: — Characteristic equation
15. The set of all matrices equivalent to a given matrix A forms a: — Class
16. The set of all matrices similar to a given matrix A forms a: — Class
17. What is the characteristic equation of a square matrix A? — det(A - λI) = 0
18. If a matrix A is similar to a diagonal matrix D, then A is called: — Diagonalizable
19. The scalar λ in the equation Av = λv is called the: — Eigenvalue
20. If two matrices are similar, they have the same: — All of the above
21. The characteristic roots of a matrix are also known as: — All of the above
22. Which of the following is NOT preserved under similarity transformations? — Rank
23. The roots of the characteristic equation of a matrix are called: — Eigenvalues
24. The canonical form under equivalence is obtained by applying: — Elementary row and column operations
25. The Cayley-Hamilton theorem can be used to: — All of the above
26. The canonical form of a matrix under equivalence transformation is unique if the matrix is: — In Smith Normal Form
27. The canonical form of a matrix under equivalence is determined by its: — Invariant factors
28. A matrix A is diagonalizable if and only if: — Its minimal polynomial has distinct roots
29. The canonical form under similarity transformation is: — Jordan Normal Form
30. If A is an n x n matrix, its characteristic polynomial has a degree of: — n
31. According to the Cayley-Hamilton theorem, the characteristic polynomial p(λ) = det(A - λI) satisfies: — p(A) = 0
32. If A is a skew-symmetric matrix with real entries, its eigenvalues are: — Purely imaginary or zero
33. Which of the following is NOT preserved under equivalence transformations? — Eigenvalues
34. If A and B are equivalent matrices, they have the same: — Rank
35. If A is a real symmetric matrix, its eigenvalues are: — Real
36. The characteristic roots of A and A⁻¹ (if A is invertible) are: — Reciprocals of each other
37. The Jordan Normal Form is a canonical form under which transformation? — Similarity
38. The canonical form under similarity is obtained by applying: — Similarity transformations
39. The set of all eigenvalues of a matrix A is called its: — Spectrum
40. If A is a diagonal matrix, its eigenvalues are: — The diagonal entries
41. What is the minimum polynomial of a matrix A? — The monic polynomial of least degree that annihilates A
42. 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: — The properties of characteristic roots
43. 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: — The properties of eigenvalues and the characteristic equation
44. The characteristic roots of A and A^T are: — The same
45. What is the trace of a square matrix? — The sum of its diagonal elements
46. The invariant factors of a matrix are: — Unique polynomials that are preserved under elementary row and column operations
47. If A is a nilpotent matrix, then all its eigenvalues are: — Zero
48. If A is a 2x2 matrix with eigenvalues λ1 and λ2, then det(A) is equal to: — λ1 * λ2
49. If A is a 2x2 matrix with eigenvalues λ1 and λ2, then tr(A) is equal to: — λ1 + λ2
50. The characteristic equation of a 2x2 matrix [[a, b], [c, d]] is: — λ² - (a+d)λ + (ad-bc) = 0
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