(2) Find the eigenvalue of A and their multiplicities.
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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---
The problem is based on a matrix \( A \) provided below. The tasks are to:
1. Compute the characteristic polynomial of the matrix \( A \).
2. Find the eigenvalues of \( A \) and their multiplicities.
3. Determine whether \( A \) is invertible and explain why.
The matrix \( A \) is:
\[
A = \begin{pmatrix}
-8 & 2 & 2 \\
2 & -5 & 4 \\
2 & 4 & -5
\end{pmatrix}
\]
**Tasks:**
1. **Compute the characteristic polynomial of the matrix \( A \)**.
To find the characteristic polynomial of a matrix \( A \), we use the formula:
\[
p(\lambda) = \det(A - \lambda I)
\]
where \( \lambda \) is a scalar, \( I \) is the identity matrix of the same dimension as \( A \), and \( \det \) denotes the determinant.
2. **Find the eigenvalues of \( A \) and their multiplicities**.
The eigenvalues of a matrix \( A \) are the roots of its characteristic polynomial.
3. **Is \( A \) invertible? Why?**
A matrix \( A \) is invertible if and only if its determinant is non-zero.
---
Explanation of processes:
- **Characteristic Polynomial**: Determine the polynomial by computing the determinant of the matrix \( (A - \lambda I) \).
- **Eigenvalues**: Solve the characteristic polynomial equation \( p(\lambda) = 0 \) to find the eigenvalues. Each root is an eigenvalue of \( A \).
- **Invertibility**: Calculate the determinant of \( A \). If the determinant is not equal to zero, the matrix \( A \) is invertible.
This matrix-related problem helps in understanding fundamental linear algebra concepts, including characteristic polynomials, eigenvalues, and matrix invertibility.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F672bf286-8abe-4b07-9ca1-0d5b2612956c%2Ffea4738c-e560-4780-8aa6-847e161ec855%2Fpqbmq58_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Below is the transcription and explanation for the given content, suitable for an educational website:
---
The problem is based on a matrix \( A \) provided below. The tasks are to:
1. Compute the characteristic polynomial of the matrix \( A \).
2. Find the eigenvalues of \( A \) and their multiplicities.
3. Determine whether \( A \) is invertible and explain why.
The matrix \( A \) is:
\[
A = \begin{pmatrix}
-8 & 2 & 2 \\
2 & -5 & 4 \\
2 & 4 & -5
\end{pmatrix}
\]
**Tasks:**
1. **Compute the characteristic polynomial of the matrix \( A \)**.
To find the characteristic polynomial of a matrix \( A \), we use the formula:
\[
p(\lambda) = \det(A - \lambda I)
\]
where \( \lambda \) is a scalar, \( I \) is the identity matrix of the same dimension as \( A \), and \( \det \) denotes the determinant.
2. **Find the eigenvalues of \( A \) and their multiplicities**.
The eigenvalues of a matrix \( A \) are the roots of its characteristic polynomial.
3. **Is \( A \) invertible? Why?**
A matrix \( A \) is invertible if and only if its determinant is non-zero.
---
Explanation of processes:
- **Characteristic Polynomial**: Determine the polynomial by computing the determinant of the matrix \( (A - \lambda I) \).
- **Eigenvalues**: Solve the characteristic polynomial equation \( p(\lambda) = 0 \) to find the eigenvalues. Each root is an eigenvalue of \( A \).
- **Invertibility**: Calculate the determinant of \( A \). If the determinant is not equal to zero, the matrix \( A \) is invertible.
This matrix-related problem helps in understanding fundamental linear algebra concepts, including characteristic polynomials, eigenvalues, and matrix invertibility.
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