4) Finde'A, if A = -2 24 -2 32 1 -2 1

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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**Problem 4: Matrix Exponential Calculation**

**Objective**: Find \( e^{tA} \) for the given matrix \( A \).

**Matrix \( A \)**:
\[
A = \begin{pmatrix} 
-2 & 2 & 4 \\ 
-2 & 3 & 2 \\ 
1 & -2 & 1 
\end{pmatrix}
\]

**Explanation**: This problem involves calculating the matrix exponential \( e^{tA} \), where \( A \) is a 3x3 matrix. The matrix exponential is an important concept in linear algebra, often used to solve systems of linear differential equations.

**Approach**:
1. **Eigenvalue Decomposition**: Find the eigenvalues and eigenvectors of the matrix \( A \).
2. **Series Expansion**: Use the series expansion for the matrix exponential \( e^{tA} = I + tA + \frac{t^2}{2!}A^2 + \frac{t^3}{3!}A^3 + \cdots \).
3. **Diagonalization**: If possible, use the diagonalization of matrix \( A \) to simplify the computation of the matrix exponential.

Understanding the matrix exponential and its applications can be beneficial for students studying advanced mathematics, physics, or engineering disciplines.
Transcribed Image Text:**Problem 4: Matrix Exponential Calculation** **Objective**: Find \( e^{tA} \) for the given matrix \( A \). **Matrix \( A \)**: \[ A = \begin{pmatrix} -2 & 2 & 4 \\ -2 & 3 & 2 \\ 1 & -2 & 1 \end{pmatrix} \] **Explanation**: This problem involves calculating the matrix exponential \( e^{tA} \), where \( A \) is a 3x3 matrix. The matrix exponential is an important concept in linear algebra, often used to solve systems of linear differential equations. **Approach**: 1. **Eigenvalue Decomposition**: Find the eigenvalues and eigenvectors of the matrix \( A \). 2. **Series Expansion**: Use the series expansion for the matrix exponential \( e^{tA} = I + tA + \frac{t^2}{2!}A^2 + \frac{t^3}{3!}A^3 + \cdots \). 3. **Diagonalization**: If possible, use the diagonalization of matrix \( A \) to simplify the computation of the matrix exponential. Understanding the matrix exponential and its applications can be beneficial for students studying advanced mathematics, physics, or engineering disciplines.
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