Algebraic operations on matrices - rank of a matrix, eigenvalues and eigenvectors, characteristic equation, Cayley–Hamilton theorem, diagonalisation and diagonalizability of unitary, orthogonal, Hermitian and symmetric matrices - Online Test

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1. What is the rank of a zero matrix of size m x n?

A) 1 B) m C) n D) 0

2. What is the maximum possible rank of an m x n matrix A?

A) m B) n C) min(m, n) D) max(m, n)

3. If A is an m x n matrix and B is an n x p matrix, what is the maximum possible rank of the product AB?

A) min(rank(A), rank(B)) B) rank(A) + rank(B) C) min(m, p) D) rank(A)

4. What is the rank of the identity matrix of size n x n?

A) 0 B) 1 C) n-1 D) n

5. If matrix A is invertible, what is its rank?

A) 0 B) 1 C) n-1 D) n

6. What is the rank of a matrix A if its determinant is non-zero?

A) Less than n B) Equal to n C) 0 D) n+1

7. Which of the following operations does NOT change the rank of a matrix?

A) Multiplying a row by a non-zero scalar B) Adding a multiple of one row to another row C) Swapping two rows D) Multiplying the matrix by another matrix

8. An eigenvalue of a matrix A is a scalar \u03BB such that Ax = \u03BBx for some non-zero vector x. What is the vector x called?

A) Eigenvector B) Eigenvalue C) Characteristic vector D) Principal vector

9. For a square matrix A, if \u03BB is an eigenvalue and v is the corresponding eigenvector, what equation must hold?

A) Av = 0 B) v = \u03BBA C) Av = \u03BBv D) A + v = \u03BB

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

A) det(A - \u03BB) = 0 B) det(A + \u03BB) = 0 C) det(\u03BBA - I) = 0 D) det(A) - \u03BB = 0

11. The roots of the characteristic equation det(A - \u03BB I) = 0 are the:

A) Eigenvectors of A B) Eigenvalues of A C) Singular values of A D) Norms of A

12. What is the Cayley-Hamilton theorem related to?

A) The rank of a matrix B) The determinant of a matrix C) A matrix satisfying its own characteristic equation D) The eigenvalues of a matrix

13. According to the Cayley-Hamilton theorem, if p(\u03BB) = det(A - \u03BB I) is the characteristic polynomial of matrix A, then:

A) p(A) = I B) p(A) = 0 C) p(A) = A D) p(A) = \u03BBI

14. If a matrix A is diagonalizable, it means that there exists an invertible matrix P such that P⁻¹AP is a:

A) Symmetric matrix B) Orthogonal matrix C) Diagonal matrix D) Unitary matrix

15. A square matrix A is diagonalizable if and only if it has:

A) n distinct eigenvalues B) n linearly independent eigenvectors C) a non-zero determinant D) a zero determinant

16. What are the eigenvalues of a diagonal matrix?

A) All zeros B) All ones C) The diagonal entries D) The sum of the diagonal entries

17. If A is a real symmetric matrix, what can be said about its eigenvalues?

A) They are always complex B) They are always purely imaginary C) They are always real D) They are always zero

18. Eigenvectors corresponding to distinct eigenvalues of a symmetric matrix are:

A) Linearly dependent B) Orthogonal C) Parallel D) Identical

19. A real matrix A is orthogonal if:

A) AᵀA = I B) AAᵀ = 0 C) AᵀA = A D) AAᵀ = I

20. What are the eigenvalues of an orthogonal matrix?

A) Always real numbers B) Always purely imaginary numbers C) Magnitudes equal to 1 D) Always zero

21. A complex matrix A is unitary if:

A) AᴴA = I B) AᵀA = I C) AᴴA = 0 D) A*A = I

22. What are the eigenvalues of a unitary matrix?

A) Always real numbers B) Always purely imaginary numbers C) Magnitudes equal to 1 D) Always zero

23. A complex matrix A is Hermitian if:

A) Aᴴ = A B) Aᵀ = A C) Aᴴ = -A D) A* = A

24. What are the eigenvalues of a Hermitian matrix?

A) Always complex B) Always purely imaginary C) Always real D) Always zero

25. If a matrix A is symmetric, then it is always:

A) Orthogonal B) Unitary C) Diagonalizable D) Invertible

26. If a matrix A is orthogonal, then it is always:

A) Symmetric B) Hermitian C) Diagonalizable D) Unitary

27. If a matrix A is unitary, then its conjugate transpose Aᴴ is equal to:

A) A B) -A C) A⁻¹ D) I

28. The sum of the eigenvalues of a matrix A is equal to its:

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

29. The product of the eigenvalues of a matrix A is equal to its:

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

30. If \u03BB is an eigenvalue of A, then k\u03BB is an eigenvalue of kA, where k is a scalar. This property relates to:

A) Rank of a matrix B) Eigenvalues and scalar multiplication C) Cayley-Hamilton theorem D) Matrix diagonalization

31. If \u03BB is an eigenvalue of A, then \u03BBⁿ is an eigenvalue of Aⁿ. This property is a consequence of:

A) Rank theorem B) Cayley-Hamilton theorem C) Eigenvalue properties D) Matrix inversion

32. If \u03BB is an eigenvalue of an invertible matrix A, then 1/\u03BB is an eigenvalue of:

A) Aᵀ B) Aⁿ C) A⁻¹ D) kA

33. For a real symmetric matrix, the eigenvectors corresponding to distinct eigenvalues are:

A) Always parallel B) Always linearly dependent C) Always orthogonal D) Never orthogonal

34. A matrix that can be diagonalized is called:

A) Singular B) Non-singular C) Diagonalizable D) Orthogonal

35. Which type of matrix is always diagonalizable?

A) Any square matrix B) A matrix with distinct eigenvalues C) A symmetric matrix D) A non-invertible matrix

36. If A = PDP⁻¹, where D is a diagonal matrix, what are the diagonal entries of D?

A) The eigenvectors of A B) The eigenvalues of A C) The columns of P D) The rows of P⁻¹

37. The columns of the matrix P in the diagonalization A = PDP⁻¹ are the:

A) Eigenvalues of A B) Eigenvectors of A C) Diagonal entries of D D) Scalar multiples of eigenvalues

38. What is the characteristic polynomial of a 2x2 identity matrix?

A) \u03BB² - 1 B) \u03BB² C) (1-\u03BB)² D) \u03BB - 1

39. What are the eigenvalues of the matrix [[2, 1], [1, 2]]?

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

40. What is the rank of the matrix [[1, 2, 3], [4, 5, 6], [7, 8, 9]]?

A) 1 B) 2 C) 3 D) 0

41. Which of the following is NOT an algebraic operation on matrices?

A) Addition B) Scalar Multiplication C) Matrix Multiplication D) Determinant calculation

42. If A is an n x n matrix and \u03BB is an eigenvalue, then det(A - \u03BB I) = 0 is satisfied for:

A) All vectors x B) Only the zero vector x C) The corresponding eigenvector(s) x D) All non-zero vectors x

43. A matrix is diagonalizable if it has n linearly independent eigenvectors. This implies that the geometric multiplicity of each eigenvalue is equal to its:

A) Algebraic multiplicity B) Eigenvalue C) Determinant D) Rank

44. What is the rank of a non-zero 1x1 matrix?

A) 0 B) 1 C) Undefined D) Depends on the element

45. If A is an m x n matrix, its row rank is equal to its:

A) Column rank B) Determinant C) Trace D) Eigenvalue

46. Which theorem states that every square matrix over a field satisfies its own characteristic polynomial?

A) Fermat's Little Theorem B) Lagrange's Theorem C) Cayley-Hamilton Theorem D) Fundamental Theorem of Algebra

47. If A is a real matrix, its eigenvalues can be:

A) Always real B) Always complex C) Real or complex conjugate pairs D) Always purely imaginary

48. For a unitary matrix U, the eigenvalues satisfy |\u03BB| = 1. This means the eigenvalues lie on the:

A) Real axis B) Imaginary axis C) Unit circle in the complex plane D) Origin

49. If A is an n x n matrix and P is an invertible matrix such that P⁻¹AP = D (diagonal), then A is said to be:

A) Singular B) Non-singular C) Diagonalizable D) Orthogonal

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

A) Spectrum B) Resolvent set C) Eigenbasis D) Characteristic set

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