Which of the following statements are always true?(1) If A is diagonalizable then cA is diagonalizable for any constant c.(i1) If the characteristic polynomial of a matrix A is equal to (1- 5) (1 – 7)°(a – 8)* then the eigenvalue 1has a geometric multiplicity of 9.(111) If A is similar to B, then A is similar to B.= 5(A) (1) and (iii) only (B) all of them (C) (i) only (D) (iii) only (E) (ii) and (iii) only (F) none of them(G) (11) only (H) (1) and (ii) only

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Answered: (1) If A is diagonalizable then cA is…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Consider the matrix A = - 13 24 8 15 (a) Calculate the det(A - xI), where x is a variable. We call the result the characteristic polynomial of A. (b) Give the two zeros 1 and 2 of this polynomial. λ1 and 22 are called the eigenvalues of the matrix A. (c) For each eigenvalue 21, find an eigenvector vi = ( ) + (8) xi # such 7 that Avi = hivi. (d) Construct the matrix D = (2). are eigenvectors v1 and v2. Calculate QDQ λ1 0 22 0 and the matrix Q whose columns -1 ; what do you notice?
in 2) Suppose that A is a square matrix and let p(x) be a polynomial with real coefficients with variable x. If p(x) = E a;x' i = 1 in , then define p (A) = E a¡A' . If, is an eigenvalue of A, the show that p (1) is an eigenvalue of p(A). i = 1
Which of the following statements are always true? (i) If A is a 3 x 3 diagonalizable matrix then A has 3 distinct eigenvalues. (ii) If the characteristic polynomial of a matrix A is equal to (2 – 3)*(2– 6)°(2 – 7)š then the eigenvalue 2 = 3 has a geometric multiplicity of 4. (iii) If A is similar to B, and B is symmetric, then A' is similar to B. A) (i) and (ii) only (B) (ii) and (iii) only (C) (ii) only (D) none of them (E) (iii) only (F) (i) only G) all of them (H) (i) and (iii) only
Please solve B,C,D,E
The characteristic polynomial of A is p(λ) = λ3−3λ2+4. Find its eigenvalues.
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If A is an (n x n)-matrix of real constants that has a complex eigenvalue and eigenvector v, then the real and imaginary parts of w(t) = elty are linearly independent real-valued solutions of x' =Ax: x₁(t) = Re(w(t)) and X₂(t) = Im(w(t)), The matrix in the following system has complex eigenvalues; use the above theorem to find the general (real-valued) solution. 0 30 -3 0 0 X 005 x(t) = = x' = Find the particular solution given the initial conditions. x(t) = = x(0) = 123
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1 1 (c) Which matrix in (A, B, C) is closest to -1 0 2 2 31. Let A = 2 0 2 2 0 (a) Find the characteristic polynomial of A. (b) Find all eigenvalues A. (c) Find the inverse of A, A-1, by using Cayley-Hamilton theorem. (d) Find the standard equation of the quadratic surface4.ry + 4rz+ 4yz 24. -1 -1 0-1 32. Let A = -1 -1 0. (a) Find the characteristic polynomial of A.

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(1) If A is diagonalizable then cA is diagonalizable for any constant c. (i1) If the characteristic polynomial of a matrix A is equal to (1 - 5) (1– 7)°(1 – 8)* then the eigenvalue A = 5 has a geometric multiplicity of 9. (ii1) If A is similar to B, then A is similar to B.