1. -2 -2 Let A = -2 -2 -2 -2 1 Determine the characteristic polynomial of A: I-1 2 XA(x) = det(rI3 – A) = det 2 I-1 2 2 I-1, b) multiplicity. Factor XA(1) and determine the eigenvalues and their algebraic c) For each eigenvalue a find a basis of the eigenspace, E.(A). d) Explain why the matrix is diagonaizable. Write down a diagonal form, D, of A and a diagonalizing matrix P such that P-1AP = D.

1. -2 -2 Let A = -2 -2 -2 -2 1 Determine the characteristic polynomial of A: I-1 2 XA(x) = det(rI3 – A) = det 2 I-1 2 2 I-1, b) multiplicity. Factor XA(1) and determine the eigenvalues and their algebraic c) For each eigenvalue a find a basis of the eigenspace, E.(A). d) Explain why the matrix is diagonaizable. Write down a diagonal form, D, of A and a diagonalizing matrix P such that P-1AP = D.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: [2 1 4. Let A = 1 1 -1 2 -1 2 Show that the characteristic polynomial fa of A is -X³ + 5A² – 21 – 8.…

Q: Compute the characteristic polynomial of 0 and hence find its eigenvalues. For each eigenvalue A…

Q: Г-4 0 21 Let A = 2 4 -8 2 0 -4 a) Find the characteristic polynomial of A. b) Find the eigenvalues…

Q: For Matrix A, find a basis for the eigenspace corresponding to λ = 3. 4 A = 3 0 0 2 2 - - 1 3 5

Q: Consider the operator R2 → R2 given by left-multiplication by the matrix: A =[−2 −1, 1 −4].…

Q: 1 3 4 1 7 9 4 2 2 (:) 1 2 A = 20 3 B = 0 2 8 C = 2 1 1 M = 2 1 1 6 6 0 0 5 1 1 a) Show that 0 is an…

Q: Find a basis B for the domain of T such that the matrix for 7 relative to B is diagonal. (Enter the…

A: The given linear transformation is .

Q: -1 -1 -1 -1 has two distinct eigenvalues A1 < d2. The matrix M : -1 -1 -1 -1 Find the eigenvalues…

A: We find eigenvalues and an orthornormal basis for each eigenspace.

Q: Let A = 9 -6 0 9 -6 0 4 5 3 eigenspace. (a) Compute the characteristic polynomial pë(λ). (b) Use…

A: Given A=9-609-60453 (a) The characteristic polynomial of A is…

Q: [20 0 -1 1 0 -2 -1] a) Find the characteristic polynomial of A. b) Find the eigenvalues of A. c) For…

Q: - Let p(t) = t5 - 6tª + 6t3 – 6t² + 5t. a) Factorise p(t) as a product of degree 1 polynomials. b)…

A: The given polynomial is p(t)=t5-6t4+6t3-6t2+5t We have to find the roots of this polynomial to write…

Q: Find a basis for the eigenspace of B below. 2 1 B = 0 3 a. B= ((0,1),(1,0)} b. B= ((- 1,0),(1,1)} B…

A: To find the Basis for eigen space , first I have to find the eigen values then find the eigen vector…

Q: 3. (a) Use change-of-basis techniques to construct a 2 x 2 matrix A that satisfies the following…

A: Given a linear transformation matrix A such that A21=321A1-2=51-2

Q: Define T: P₂ → P₂ by Find the eigenvalues. (Enter your answers from smallest to largest.) (21, 22,…

Q: 4 1 3 0 6. Let A = -1 2 0 4 1 2 (a) calculate determinants.) Find the characteristic polynomial of…

Q: Let A ght Texas St 2 0 0 red. 3 b -1 10 1 Answer the following

A: Given information: A=2003b-1101 and the value of b is given as 3. To find: a) Characteristic…

Q: 2. Let A = = 1 -4 -2 (a) Compute the eigenvalues of A. (b) Find a basis for each eigenspace of A.…

Q: 2. Let T = 1 0 2-3 0 3 and denote I3 = 0 1 0 001 (c) Express an arbitrary vector v = nation of…

Q: , a linear operator and a scalar λ are given. Show that is an eigenvalue of the operator and…

Q: For each Linear Operator: S (x, y) = (2x + y, x + 2y) T(x,y,z) = (z,y,x) to determine: (a) The…

A: Note: Since we can solve at most three subparts in a question, we have solved the first three…

Q: 3. 401 maldoq sol -2 10 Three linearly independent eigen- -2 0 1 vectors of A are v₁ = (-1, 1, 1),…

Q: B = 5 3 -3 -2 1\ 0 1 2 1

Q: 4 2 2 3 basis for eigenspace for -1 1-3 that corresponds to λ = 3.

A: Eigen vector equation corresponding to eigen value λ is A-λIX=0 We have A=423-11-3249 and λ=3

Q: For the given matrix A. 54 A = 4 5 2 22 (a) Find the characteristic polynomial in expanded form. (b)…

Q: Let T: R5 → R5 be a linear operator whose characteristic polynomial is pT(x) = (x−3)3(x−5)2. Let V3…

Q: Find the eigenvalues and a basis for each of the eigenspaces of the linear operator T: P2 - P2…

Q: 1 0 0 Let A = -3 1 3 3 0-2 (a) Find all eigenvalues of A. (b) For each eigenvalue A of A, find a…

A: NOTE: As per the guidelines, repost part (d).

Q: 1. Let A = 1 2 4 1 0 0 0-1 3 Answer the following a. Find the characteristic polynomial of A, and…

Q: 1 -1 A: 1 -1 2 1 -1 1 Find all complex eigenvalues of A and find a basis for each complex eigenspace…

A: The eigenvalues and eigenvectors of matrix are given below :

Q: 6. M = 2 -7 1-6 a. Find the characteristic polynomial and eigenvalues of M. b. Find a basis for the…

Q: Question 2 Give a matrix A as follows -2-6 0 A = 2 7 0 1 2 -1 a) Find the polynomial characteristics…

Q: 1. a) For the matrix A below, find the eigenvalues, the eigenvectors, a basis B for which the matrix…

Q: (a) Let M be a square matrix such that M² = 0. Let t, c be scalars and c‡0. Show that

A: We shall solve first question only as you have asked more than one different question - for others…

Q: Suppose TEL(C³) is defined by T(21, 22, 23) = (621 +372 +423,622 +223, 723) (a) Write down the…

Question

1.
-2 -2
Let A =
-2
1
-2
-2 -2
1.
a)
Determine the characteristic polynomial of A:
2
2
XA(1) = det(rl3 – A) = det
2
I - 1
2
2
2
I - 1
b)
multiplicity.
Factor XA(r) and determine the eigenvalues and their algebraic
c)
For each eigenvalue a find a basis of the eigenspace, E.(A).
d)
Explain why the matrix is diagonaizable.
e)
P such that P-1AP = D.
Write down a diagonal form, D, of A and a diagonalizing matrix

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

Step 2

Step 3

Step 4

Step 5

Step by step

Solved in 5 steps with 5 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Matrix Eigenvalues and Eigenvectors$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

For the system of differential equations, []+[4] y 6 8 a) Find the characteristic polynomial of the matrix of coefficients A. CA(X) A1, A2 = U₁ = = b) Find the eigenvalues of A. Enter the eigenvalues as a list in ascending order separated by commas. U₂ = c) Find the eigenvectors assuming u₁ is the eigenvector associated with the smaller eigenvalue X₁ and u₂ is the eigenvector associated with the larger eigenvalue A₂. Enter the eigenvectors as a matrix with an appropriate size. v(t) y' = d) Determine a general solution to the system by completing the following steps. i. Find v(t) = = [y−¹(t)f(t)dt . yp(t) = -7 y (t) ii. Find a particular solution y(t). = ii. Then a general solution for the system in the matrix form is
2 - 1) Find the eigenvalues and eigenvectors of the matrix B = -1 1 - 2 1 -1 2) Given B = {B,, B2, ß3} where Bi B2 =|0, B3 = - 3 %3D 1 i) Show that B is a basis for R³. ii) Determine an orthonormal basis of R based on B.
Consider the following matrices A and B and vector b, attached in image. (a) Find the characteristic polynomial of matrix A.(b) Find eigenvalues of the matrix A.(c) Find a basis for the eigenspaces of matrix A.(d) What is the algebraic and geometric multiplicities of the eigenvalues of A.
a a Consider the function T : R³ → R³ given by T а — b 2а + 3b + с The eigenvalues of T are A= 1 and A = -1 Find a basis for E2(T) and the geometric multiplicity of d = 1 and A= -1
Let T:P2 → P2 be defined by T(a + bx + cx²) = (5a + 6b + 2c) – (b + 8c)x + (a – 2c)x². Find bases for the eigenspaces of T.
[4 -1 6] Given that 2 is an eigenvalue for A = |2 [2 1 Find a basis of its -1 8 Eigen space.
4. Let V = C and define 0 : V → V by •(: )-( -15a + 24b – 48c — 6а + 10b — 18с 2а — 36 + 7с Compute the characteristic polynomial of 0 and hence find its eigenvalues. For each eigenvalue A compute the eigenspace Ex and write down a basis of it. Hence decide whether there exists a basis of V consisting of eigenvectors for 0.
Let A = (a) By calculating and factorizing C₁(2), the characteristic polynomial of A, show that -1 is the only eigenvalue of A, and then find a corre- sponding eigenvector u. -9 2 (b) Find a vector v R2 such that (A + I)v = u, and show by direct calculation that P-¹AP = B where (c) Using the fact that P = [u v] and B = Bn PA 0 = (-1)" [ 7] for every integer n, find An explicitly in terms of n. Hint: Begin by using the equality A = PBP-¹ to express A" in terms of P and B.
Please help with parts d,e,f with solution and steps!
Vv