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Problem 1. Let A = 2 2 2-5 4 4-5/ 2 (1) Compute the characteristic polynomial of the matrix A. (2) Find the eigenvalue of A and their multiplicities. (3) Is A invertible? Why?

Problem 1. Let A = 2 2 2-5 4 4-5/ 2 (1) Compute the characteristic polynomial of the matrix A. (2) Find the eigenvalue of A and their multiplicities. (3) Is A invertible? Why?

Advanced Engineering Mathematics
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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**Problem 5.** Find the orthogonal projections of the \( b_0 \) of Problem 4 onto the various eigenspaces of the matrix \( A \) of Problem 1.
Transcribed Image Text:**Problem 5.** Find the orthogonal projections of the \( b_0 \) of Problem 4 onto the various eigenspaces of the matrix \( A \) of Problem 1.
**Problem 1** Let \( A = \begin{pmatrix} -8 & 2 & 2 \\ 2 & -5 & 4 \\ 2 & 4 & -5 \end{pmatrix} \). 1. Compute the characteristic polynomial of the matrix \( A \). 2. Find the eigenvalue of \( A \) and their multiplicities. 3. Is \( A \) invertible? Why? **Problem 2** Find a basis for each eigenspace of the matrix \( A \) of Problem 1. **Problem 3** Find an orthogonal diagonalization of the matrix \( A \) of Problem 1. **Problem 4** Define a dynamical system with \( b_0 = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \) and \( b_{n+1} = Ab_n \), for \( n = 1, 2, 3, \ldots \), with \( A \) as in Problem 1. Find a formula for \( b_n \) for each \( n \).
Transcribed Image Text:**Problem 1** Let \( A = \begin{pmatrix} -8 & 2 & 2 \\ 2 & -5 & 4 \\ 2 & 4 & -5 \end{pmatrix} \). 1. Compute the characteristic polynomial of the matrix \( A \). 2. Find the eigenvalue of \( A \) and their multiplicities. 3. Is \( A \) invertible? Why? **Problem 2** Find a basis for each eigenspace of the matrix \( A \) of Problem 1. **Problem 3** Find an orthogonal diagonalization of the matrix \( A \) of Problem 1. **Problem 4** Define a dynamical system with \( b_0 = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \) and \( b_{n+1} = Ab_n \), for \( n = 1, 2, 3, \ldots \), with \( A \) as in Problem 1. Find a formula for \( b_n \) for each \( n \).
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