# IIT JAM Mathematics Test Series 2023 : Linear Algebra – Diagonalizability and Canonical forms

Practice IIT JAM Mathematics Test Series for Free only On www.examflame.com and take your preparation to the another level. In the Series of Tests for IIT JAM 2023 this is the test of topic Diagonalizability and Canonical forms of the Chapter Linear Algebra. These Questions are prepared as per the Latest Syllabus of IIT JAM Mathematics 2023 . Practicing mock tests/Test series, you will get an idea about how and which type of Question will ask in the Examination. It also boost your confidence level. Also Solve IIT JAM Mathematics Previous Year Question Paper. And Don’t Forget to Shore with Your Friends.

## IIT JAM Mathematics Test Series 2023 : Linear Algebra – Diagonalizability and Canonical forms

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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