1. ( Let A = Eigenvalues: Eigenvectors: 1 0 0 0 2 2 0 0 7120 Is the matrix A diagonalizable? 3382 0 Find the eigenvalues and eigenvectors of the matrix A. If it is diagonalizable find a matrix P and a diagonal matrix D such that A = PDP-¹.

1. ( Let A = Eigenvalues: Eigenvectors: 1 0 0 0 2 2 0 0 7120 Is the matrix A diagonalizable? 3382 0 Find the eigenvalues and eigenvectors of the matrix A. If it is diagonalizable find a matrix P and a diagonal matrix D such that A = PDP-¹.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.7: The Inverse Of A Matrix

Problem 30E

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Question

I'm currently struggling to exclusively solve this problem using matrix notation, and I'm seeking your assistance. The requirement is to find a solution using only matrix notation, without employing any other methods. Could you kindly guide me through a detailed, step-by-step explanation in matrix notation to help me solve the problem and arrive at the final solution?

Can you please label the parts so I can follow along and can you please do it step by step using the matrix form

### Problem Statement

Given the matrix

\[ A = \begin{pmatrix}
1 & 2 & -1 & 3 \\
0 & 2 & 1 & 3 \\
0 & 0 & 2 & 0 \\
0 & 0 & 0 & 2
\end{pmatrix} \]

1. Find the eigenvalues and eigenvectors of the matrix \( A \).

#### Eigenvalues:
_________________________

#### Eigenvectors:
_________________________

2. Is the matrix \( A \) diagonalizable?

3. If it is diagonalizable, find a matrix \( P \) and a diagonal matrix \( D \) such that \( A = PDP^{-1} \).

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Step 1: Finding eigenvalues of A

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Step 2: Finding eigenvectors of A

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Step 3: Finding eigenvectors of A

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Step 4: Finding eigenvectors of A

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I'm having a hard time understanding how they're getting these vectors from the matrix. I follow up to plugging in the eigenvalues and solving the variables, then I get lost for like the first one beta = alpha = 0. When I solve and plug in beta = 2*alpha = 2 * (2*beta) -> beta = 4beta, so is it zero because thats the only number that makes this equality? So like it's not so much algebra after solving as looking at the equality and making sense of it?