Questions on Diagonalizability and Canonical forms with Answers

Questions on Diagonalizability and Canonical forms with Answers

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Questions on Diagonalizability and Canonical forms with Answers


Q1. An n×n complex matrix A satisfies Ak =In , 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 + A2 +…+ Ak = 0, The n×n zero matrixes.
(c) tr(A) + tr(A2) +…+ tr(Ak-1) = -n
(d) A-1 + A-2 + …+ A-(k-l) = -In


Q2. Let S:Rn → Rn be given by S(v) = αv for a fixed α∈R, α ≠ 0. Let T: Rn → Rn be a linear transformation such that B = {v1,…,vn} 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 ∈ R4 , 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) A2 = A
(c) Trace (A) = n – 2
(d) A2 = 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 A3 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) Ak = 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 A2 = 0, then A is diagonalizable over complex numbers.
(b) If A2 = I, then A is diagonalizable over real numbers.
(c) If A2 = 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 R4.
(b) Null space of A contains only the 0 vector.
(c) A has 4 distinct eigenvalues.
(d) Image of the linear transformation x → Ax on R4 is R4


Q14. Let A∈M10(C), the vector space of 10×10 matrices with entries in C. Let WA be the subspace of M10(C) spanned by {An | n ≥ 0} .Choose the correct statements

(a) For any A, dim (WA) ≤ 10
(b) For any A, dim (WA) < 10
(c) For some A, 10 < dim (WA) < 100
(d) For some A, dim(WA) = 100


Q15. Let A be a complex 3×3 matrix with A3 = -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 Mn(R) denote the vector space of n×n real matrices. Let B ∈ Mn(R) be an orthogonal matrix and let Bt denote the transpose of B Consider WB ={BtAB : A ∈ Mn(R)}. Which of the following are necessarily true?

(a) WB is a subspace of Mn(R) and dim WB ≤ rank (B)
(b) WB is a subspace of Mn(R) and dimWB = rank (B) rank (B’)
(c) WB = Mn(R)
(d) WB is not a subspace of Mn(R).


Q18. Let A be a non-zero linear transformation on a real vector space V of dimension n. Let the subspace Vo⊂ V be the image of V under A . Let k = dim Vo < n and suppose that for some λ ⊂ R. A2 = λA. Then

(a) λ =1
(b) Det A = |λ|n
(c) λ is the only eigenvalue of A.
(d) There is a nontrivial subspace V1 ⊂ V such that Ax = 0 for all x∈V1


Q19. Let C be a n×n real matrix. Let W be the vector space spanned by {I,C,C2,…,C2n} the dimension of the vector space W is

(a) 2n
(b) At most n
(c) n2
(d) At most 2n


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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)

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