Matrix A is factored in the form PDP-1. Use the Diagonalization Theorem to find the eigenvalues of A and a basis for each eigenspace. 2 2 2 5 0 **** 2 0-1 010 2-2 0 001 A = 221 13 1 122 1 1 1 8 4 8 1 8 1 4 3 8 1 4 Select the correct choice below and fill in the answer boxes to complete your choice. (Use a comma to separate vectors as needed.) O A. There is one distinct eigenvalue, λ = OB. In ascending order, the two distinct eigenvalues are ₁ = OC. In ascending order, the three distinct eigenvalues are >₁ {}, respectively. = A basis for the corresponding eigenspace is {}. and ₂ = ₂^₂= {}and {}, respectively. Bases for the corresponding eigenspaces are {}, {}, and Bases for the corresponding eigenspaces are , and 23 =
Matrix A is factored in the form PDP-1. Use the Diagonalization Theorem to find the eigenvalues of A and a basis for each eigenspace. 2 2 2 5 0 **** 2 0-1 010 2-2 0 001 A = 221 13 1 122 1 1 1 8 4 8 1 8 1 4 3 8 1 4 Select the correct choice below and fill in the answer boxes to complete your choice. (Use a comma to separate vectors as needed.) O A. There is one distinct eigenvalue, λ = OB. In ascending order, the two distinct eigenvalues are ₁ = OC. In ascending order, the three distinct eigenvalues are >₁ {}, respectively. = A basis for the corresponding eigenspace is {}. and ₂ = ₂^₂= {}and {}, respectively. Bases for the corresponding eigenspaces are {}, {}, and Bases for the corresponding eigenspaces are , and 23 =
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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![### Eigenvalues and Eigenvectors of Matrix \( A \)
Consider Matrix \( A \), which is factored in the form \( PDP^{-1} \). Use the Diagonalization Theorem to find the eigenvalues of \( A \) and a basis for each eigenspace.
\[
A = \begin{bmatrix}
2 & 2 & 1 \\
1 & 3 & 1 \\
1 & 2 & 2
\end{bmatrix} = \begin{bmatrix}
2 & 2 & 1 \\
2 & 0 & -1 \\
2 & -2 & 0
\end{bmatrix}
\begin{bmatrix}
5 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
\frac{1}{8} & \frac{1}{4} & \frac{1}{8} \\
\frac{1}{8} & \frac{1}{4} & -\frac{3}{8} \\
\frac{1}{4} & -\frac{1}{2} & \frac{1}{4}
\end{bmatrix}
\]
### Multiple Choice Question
**Instructions:**
Select the correct choice below and fill in the answer boxes to complete your choice.
(Use a comma to separate vectors as needed.)
- **A.** There is one distinct eigenvalue, \( \lambda = \ \_\_\_\_\_\_ \). A basis for the corresponding eigenspace is \( \{ \ \_\_\_\_\_\_ \} \).
- **B.** In ascending order, the two distinct eigenvalues are \( \lambda_1 = \ \_\_\_\_\_\_ \) and \( \lambda_2 = \ \_\_\_\_\_\_ \). Bases for the corresponding eigenspaces are \( \{ \ \_\_\_\_\_\_ \} \) and \( \{ \ \_\_\_\_\_\_ \} \), respectively.
- **C.** In ascending order, the three distinct eigenvalues are \( \lambda_1 = \ \_\_\_\_\_\_ \), \( \lambda_2 = \ \_\_\_\_\_\_ \), and \( \lambda_3 = \ \_\_\_\_\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff893ad80-e3fa-4d69-87b2-5d427e54a923%2F489b049f-aa8e-4e69-922c-0578d6cc30a7%2Fhagg0d9_processed.png&w=3840&q=75)
Transcribed Image Text:### Eigenvalues and Eigenvectors of Matrix \( A \)
Consider Matrix \( A \), which is factored in the form \( PDP^{-1} \). Use the Diagonalization Theorem to find the eigenvalues of \( A \) and a basis for each eigenspace.
\[
A = \begin{bmatrix}
2 & 2 & 1 \\
1 & 3 & 1 \\
1 & 2 & 2
\end{bmatrix} = \begin{bmatrix}
2 & 2 & 1 \\
2 & 0 & -1 \\
2 & -2 & 0
\end{bmatrix}
\begin{bmatrix}
5 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
\frac{1}{8} & \frac{1}{4} & \frac{1}{8} \\
\frac{1}{8} & \frac{1}{4} & -\frac{3}{8} \\
\frac{1}{4} & -\frac{1}{2} & \frac{1}{4}
\end{bmatrix}
\]
### Multiple Choice Question
**Instructions:**
Select the correct choice below and fill in the answer boxes to complete your choice.
(Use a comma to separate vectors as needed.)
- **A.** There is one distinct eigenvalue, \( \lambda = \ \_\_\_\_\_\_ \). A basis for the corresponding eigenspace is \( \{ \ \_\_\_\_\_\_ \} \).
- **B.** In ascending order, the two distinct eigenvalues are \( \lambda_1 = \ \_\_\_\_\_\_ \) and \( \lambda_2 = \ \_\_\_\_\_\_ \). Bases for the corresponding eigenspaces are \( \{ \ \_\_\_\_\_\_ \} \) and \( \{ \ \_\_\_\_\_\_ \} \), respectively.
- **C.** In ascending order, the three distinct eigenvalues are \( \lambda_1 = \ \_\_\_\_\_\_ \), \( \lambda_2 = \ \_\_\_\_\_\_ \), and \( \lambda_3 = \ \_\_\_\_\
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