to show why (e^(At))=the shown matrix, write the matrix exponential (as an infinite sum) for the system of differential equations given in matrix form for the first coefficient matrix. **Matrix Exponential Example**In this task, we explore the matrix exponential to demonstrate why $ e^{At} $ results in a specific form. Given a matrix $ A $:\[ A = \begin{bmatrix} 1 & 0 \\ 0 & 3 \end{bmatrix} \]We aim to show that:\[ e^{At} = \begin{bmatrix} e^t & 0 \\ 0 & e^{3t} \end{bmatrix} \]### ExplanationThe exponential of a matrix $ A $, denoted as $ e^{At} $, is defined as the power series:\[ e^{At} = I + At + \frac{(At)^2}{2!} + \frac{(At)^3}{3!} + \cdots \]For a diagonal matrix $ A $, such as the one given, the matrix exponential can be computed by exponentiating each of the diagonal elements separately.**Matrix Exponential Calculation:**1. **Diagonal Elements**: - The element in the first row and column of matrix $ A $ is 1, leading to $ e^{1 \cdot t} = e^t $. - The element in the second row and column is 3, leading to $ e^{3 \cdot t} = e^{3t} $.2. **Off-Diagonal Elements**: - The off-diagonal elements remain 0 since matrix $ A $ has 0's in those positions.Thus, by exponentiating the diagonal elements independently, the resulting matrix exponential $ e^{At} $ is:\[ e^{At} = \begin{bmatrix} e^t & 0 \\ 0 & e^{3t} \end{bmatrix} \]This concise representation demonstrates the power of computing the matrix exponential for diagonal matrices.

Answered: write the Matrix exponential For to…

Math

Advanced Math

write the Matrix exponential For to show why ett - fet o #* = [6* > ] O A = [1 2] 3

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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A: see below the answer

Question

to show why (e^(At))=the shown matrix, write the matrix exponential (as an infinite sum) for the system of differential equations given in matrix form for the first coefficient matrix.

**Matrix Exponential Example**

In this task, we explore the matrix exponential to demonstrate why $ e^{At} $ results in a specific form.

Given a matrix $ A $:

\[ A = \begin{bmatrix} 1 & 0 \\ 0 & 3 \end{bmatrix} \]

We aim to show that:

\[ e^{At} = \begin{bmatrix} e^t & 0 \\ 0 & e^{3t} \end{bmatrix} \]

### Explanation

The exponential of a matrix $ A $, denoted as $ e^{At} $, is defined as the power series:

\[ e^{At} = I + At + \frac{(At)^2}{2!} + \frac{(At)^3}{3!} + \cdots \]

For a diagonal matrix $ A $, such as the one given, the matrix exponential can be computed by exponentiating each of the diagonal elements separately.

**Matrix Exponential Calculation:**

1. **Diagonal Elements**:
- The element in the first row and column of matrix $ A $ is 1, leading to $ e^{1 \cdot t} = e^t $.
- The element in the second row and column is 3, leading to $ e^{3 \cdot t} = e^{3t} $.

2. **Off-Diagonal Elements**:
- The off-diagonal elements remain 0 since matrix $ A $ has 0's in those positions.

Thus, by exponentiating the diagonal elements independently, the resulting matrix exponential $ e^{At} $ is:

\[ e^{At} = \begin{bmatrix} e^t & 0 \\ 0 & e^{3t} \end{bmatrix} \]

This concise representation demonstrates the power of computing the matrix exponential for diagonal matrices.

With integration, one of the major concepts of calculus. Differentiation is the derivative or rate of change of a function with respect to the independent variable.

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