Questions on Diagonalizability and Canonical forms with Answers

**Questions on Diagonalizability and Canonical forms with Answers**

**Q1. An n×n complex matrix A satisfies A ^{k} =I_{n} , the n×n identity matrix, where k is a positive integer >1. Suppose I is not an eigenvałue of A. Then which of the following statements are necessarily true?**

(a) A is diagonalizable.

(b) A + A^{2} +…+ A^{k} = 0, The n×n zero matrixes.

(c) tr(A) + tr(A^{2}) +…+ tr(A^{k-1}) = -n

(d) A^{-1} + A^{-2} + …+ A^{-(k-l)} = -I_{n}

**Q2. Let S:R ^{n} → R^{n} be given by S(v) = αv for a fixed α∈R, α ≠ 0. Let T: R^{n} → R^{n} be a linear transformation such that**

**B = {v**

_{1},…,v_{n}} is a set of linearly independent eigenvectors of T. Then(a) The matrix of T with respect to B is diagonal.

(b) The matrix of (T – S) with respect to B is diagonal.

(c) The matrix of T with respect to B is not necessarily diagonal, but is upper triangular.

(d) The matrix of T with respect to B is diagonal, but the matrix of (T- S) with respect to B is not diagonal.

**Q3. Let A be an invertible 4×4 real matrix. Which of the following are NOT true?**

(a) Rank A = 4.

(b) For every vector b ∈ R^{4} , Ax = b has exactly one solution.

(c) dim(nullspace A) ≥ 1.

(d) 0 is an eigenvalue of A.

**Q4. Let u be a real n×l vector satisfying u’ u = 1, where u’ the transpose of u . Define A = I – 2uu’ where I is the nth order identity matrix. Which of the following statements are true?**

(a) A is singular.

(b) A^{2} = A

(c) Trace (A) = n – 2

(d) A^{2} = I

**Q5. Let P be a 2×2 complex matrix such that p*p is the identity matrix, where p* is the conjugate transpose of P. Then the Eigen values of P are**

(a) real

(b) complex conjugates of each other

(c) reciprocals of each other

(d) of modulus 1

**Q9. Let J denote a 101×101 matrix with all the entries equal to 1 and let I denote the identity matrix of order 101. Then the determinant of J – I is**

(a) 101

(b) 1

(c) 0

(d) 100

**Q10. Let A be a 5×5 matrix with real entries such that the sum of the entries in each row of A is 1. Then the sum of all the entries in A ^{3} is**

(a) 3

(b) 15

(c) 5

(d) 125

**Q11. For a fixed positive integer n 2 ≥ 3 , let A be the n ×n matrix defined as A = I – ^{1}/_{n} J, where I is the identity matrix and J is the n×n matrix with all entries equal to 1. Which of the following statements is NOT true?**

(a) A^{k} = A for every positive integer k.

(b) Trace (A) = n-1

(c) Rank (A) + Rank (I – A) = n.

(d) A is invertible.

**Q12. Let A be an n×n matrix with real entries. Which of the following is correct?**

(a) If A^{2} = 0, then A is diagonalizable over complex numbers.

(b) If A^{2} = I, then A is diagonalizable over real numbers.

(c) If A^{2} = A, then A is diagonalizable only over complex numbers.

(d) The only matrix of size n satisfying the characteristic polynomial of A is A.

**Q13. Let A be a 4×4 invertible real matrix. Which of the following is NOT necessarily true?**

(a) The rows of A form a basis of R^{4}.

(b) Null space of A contains only the 0 vector.

(c) A has 4 distinct eigenvalues.

(d) Image of the linear transformation x → A_{x} on R^{4} is R^{4}

**Q14. Let A∈M _{10}(C), the vector space of 10×10 matrices with entries in C. Let W_{A} be the subspace of M_{10}(C) spanned by {A_{n} | n ≥ 0} .Choose the correct statements**

(a) For any A, dim (W_{A}) ≤ 10

(b) For any A, dim (W_{A}) < 10

(c) For some A, 10 < dim (W_{A}) < 100

(d) For some A, dim(W_{A}) = 100

**Q15. Let A be a complex 3×3 matrix with A ^{3} = -I. Which of the following statements are correct?**

(a) A has three distinct eigenvalues

(b) A is diagonalizable over C

(c) A is triangularizable over C

(d) A is non-singular

**Q17. Let n be an integer ≥ 2 and let M _{n}(R) denote the vector space of n×n real matrices. Let B ∈ M_{n}(R) be an orthogonal**

**matrix and let B**

^{t}denote the transpose of B Consider W_{B}={B^{t}AB : A ∈ M_{n}(R)}. Which of the following are necessarily true?(a) W_{B} is a subspace of M_{n}(R) and dim W_{B} ≤ rank (B)

(b) W_{B} is a subspace of M_{n}(R) and dimW_{B} = rank (B) rank (B’)

(c) W_{B} = M_{n}(R)

(d) W_{B} is not a subspace of M_{n}(R).

**Q18. Let A be a non-zero linear transformation on a real vector space V of dimension n. Let the subspace V _{o}⊂ V be the image of V under A . Let k = dim V_{o} < n and suppose that for some λ ⊂ R. A^{2} = λA. Then**

(a) λ =1

(b) Det A = |λ|^{n}

(c) λ is the only eigenvalue of A.

(d) There is a nontrivial subspace V_{1} ⊂ V such that A_{x} = 0 for all x∈V_{1}

**Q19. Let C be a n×n real matrix. Let W be the vector space spanned by {I,C,C ^{2},…,C^{2n}} the dimension of the vector space W is**

(a) 2n

(b) At most n

(c) n^{2}

(d) At most 2n

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*Answer Key*

*Answer Key*

01. | (a), (c), (d) | 06. | (d) | 11. | (a) |

02. | (a), (b) | 07. | (c) | 12. | (b), (c), (d) |

03. | (c), (d) | 08. | (d) | 13. | (b), (c) |

04. | (c), (d) | 09. | (b) | 14. | (d) |

05. | (d) | 10. | (c) | 15. | (b) |