Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Problem Statement:**
Find the matrix \( A \) that has the given eigenvalues and corresponding eigenvectors.
**Given:**
Eigenvalue \( \lambda_1 = 3 \) with the eigenvector \(\begin{bmatrix} 4 \\ 11 \end{bmatrix}\),
Eigenvalue \( \lambda_2 = 1 \) with the eigenvector \(\begin{bmatrix} 1 \\ 3 \end{bmatrix}\).
**Matrix Form:**
\[ A = \begin{bmatrix} \quad & \quad \\ \quad & \quad \end{bmatrix} \]
**Explanation:**
To find matrix \( A \), you will use the given eigenvalues and eigenvectors. A matrix can be constructed by using the formula:
\[ A = PDP^{-1} \]
where \( P \) is a matrix formed by placing the eigenvectors as columns, and \( D \) is a diagonal matrix formed by placing the eigenvalues along the diagonal.](https://content.bartleby.com/qna-images/question/ef36e706-47e8-4994-a8d4-317113440949/aaad45f6-bee2-48bc-9c4d-89fd8cef101e/4g3gof_processed.jpeg)
Transcribed Image Text:**Problem Statement:**
Find the matrix \( A \) that has the given eigenvalues and corresponding eigenvectors.
**Given:**
Eigenvalue \( \lambda_1 = 3 \) with the eigenvector \(\begin{bmatrix} 4 \\ 11 \end{bmatrix}\),
Eigenvalue \( \lambda_2 = 1 \) with the eigenvector \(\begin{bmatrix} 1 \\ 3 \end{bmatrix}\).
**Matrix Form:**
\[ A = \begin{bmatrix} \quad & \quad \\ \quad & \quad \end{bmatrix} \]
**Explanation:**
To find matrix \( A \), you will use the given eigenvalues and eigenvectors. A matrix can be constructed by using the formula:
\[ A = PDP^{-1} \]
where \( P \) is a matrix formed by placing the eigenvectors as columns, and \( D \) is a diagonal matrix formed by placing the eigenvalues along the diagonal.
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