Find e for A = 1 -1 1 2 34 002
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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Question
![### Problem Statement
**2. Find \( e^A \) for**
\[
A = \begin{bmatrix}
1 & 1 & 2 \\
-1 & 3 & 4 \\
0 & 0 & 2
\end{bmatrix}
\]
### Explanation
This problem requires finding the matrix exponential \( e^A \) for a given matrix \( A \). The matrix \( A \) is a 3x3 matrix with the following entries:
\[
A = \begin{bmatrix}
1 & 1 & 2 \\
-1 & 3 & 4 \\
0 & 0 & 2
\end{bmatrix}
\]
### Key Concepts
- **Matrix Exponential:** The matrix exponential \( e^A \) for any square matrix \( A \) is defined as:
\[
e^A = \sum_{k=0}^{\infty} \frac{A^k}{k!}
\]
This expression involves an infinite series, where \( A^k \) indicates \( A \) raised to the power \( k \) and \( k! \) denotes the factorial of \( k \).
- **Diagonalization (if possible):** If \( A \) is diagonalizable, it can be expressed as \( A = PDP^{-1} \), where \( D \) is a diagonal matrix. Then, \( e^A \) can be computed as \( e^A = Pe^DP^{-1} \), with \( e^D \) being a diagonal matrix where each diagonal element is the exponential of the corresponding diagonal element of \( D \).
- **Jordan Form (if applicable):** For matrices that are not diagonalizable, the Jordan canonical form may be used. \( e^A \) can then be found through a similar process involving the Jordan form.
This concept is widely used in solving systems of differential equations, among other applications in linear algebra and applied mathematics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8f24dc5d-a43a-4abd-9b6e-c5e020f186b5%2F5bc4d0e0-8bb9-4dea-8c1e-e573d615d32e%2Fel94zac_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**2. Find \( e^A \) for**
\[
A = \begin{bmatrix}
1 & 1 & 2 \\
-1 & 3 & 4 \\
0 & 0 & 2
\end{bmatrix}
\]
### Explanation
This problem requires finding the matrix exponential \( e^A \) for a given matrix \( A \). The matrix \( A \) is a 3x3 matrix with the following entries:
\[
A = \begin{bmatrix}
1 & 1 & 2 \\
-1 & 3 & 4 \\
0 & 0 & 2
\end{bmatrix}
\]
### Key Concepts
- **Matrix Exponential:** The matrix exponential \( e^A \) for any square matrix \( A \) is defined as:
\[
e^A = \sum_{k=0}^{\infty} \frac{A^k}{k!}
\]
This expression involves an infinite series, where \( A^k \) indicates \( A \) raised to the power \( k \) and \( k! \) denotes the factorial of \( k \).
- **Diagonalization (if possible):** If \( A \) is diagonalizable, it can be expressed as \( A = PDP^{-1} \), where \( D \) is a diagonal matrix. Then, \( e^A \) can be computed as \( e^A = Pe^DP^{-1} \), with \( e^D \) being a diagonal matrix where each diagonal element is the exponential of the corresponding diagonal element of \( D \).
- **Jordan Form (if applicable):** For matrices that are not diagonalizable, the Jordan canonical form may be used. \( e^A \) can then be found through a similar process involving the Jordan form.
This concept is widely used in solving systems of differential equations, among other applications in linear algebra and applied mathematics.
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