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

See similar textbooks

Related questions

Q: The Quadratic equation 2x^(2)+6x+7=0 has roots \alpha and B. Calculate the value of (1)/(\alpha…

Q: don't know why you transfer Z^k to 2^k

Q: Find e^(tA). Use e^(tA) = P e^(tD) P^(-1). The matrix is \begin{pmatrix}1&2&0\\…

Q: The eigenvalues of the coefficient matrix can be found by inspection or factoring. Apply the…

Q: For A S calculate I+tA+ 0.904837 (-2-6) X 10 -0.200334 X 1.10517 t2 The actual value of e014, to six…

Q: Determine the eigenvalues and eigenvectors of the matrix 3 -2 01 =-2 3 0 005]

A: The given matrix,C=3-20-230005

Q: 1+t t -t ₁ -²₁ ) = e²₁ et4 1 1-t

A: Observed that Given that By comparing coefficients of above two expressions we get ,,But noted that…

Q: Let's now find the corresponding eigenvectors: Plugging in a₁: Plugging in a2: 0 1 (9) 0 - 0 - λ 1 0…

A: The given matrix is 010100002. To Find: Corresponding eigenvectors.

Q: ithout solving it, rewrite the given second order differential equation t'u" - 3.5tu' + (t² − 1)u =…

Q: If matrix A, B and C are invertible matrix of same order then (ABC)^(-1) = * C^-1) x B^(-1) x A^(-1)…

A: Given that A,B and C are invertible matrices of the same order.

Q: Diaganolize the following matrix such that: D=(P^T)(A)(P). Rather than finding the inverse matrix,…

Q: Which of the following subsets of the sets in the set of real square matrices of size 2 x 2 is…

Q: Hence, write down the associated modal matrix P and diagonal spectrum matrix D, and use these to…

Q: Find the first three iterations (by hand) obtained by the Symmetric Power method applied to the…

Q: dz dt 01 1 = [1] ² + HH] z 10 1

Q: 8. Solve the matrix equation Ax = b where A 1 0 and b %3D -4

A: The given matrix equation is - A = 121130-1-41 and b = 11-1 Converting the given matrix equation…

Q: -15 12 da dt -18 15 with the initial value -12 (0)D, -16 8e^(-3t)+4e^(-3t) a (t) = 12e^(3t)+4e^(3t)

Q: Please explain how the chracteristic polynomial = -(lambda - 3)(lambda - 3)(lambda - 2) for the…

Q: — 2х — 4у Find the solution to the linear system of differential equations satisfying the initial…

Q: If A is a 3x3 matrix with eigenvalues 1 and 2, which option is impossible for diagonal matrix…

Q: Vs = ft - f- f+ + f =

Q: A,B square matrices A^3=B^3, A^2B=B^2A, A^2+B^2 invertible Show A=B

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.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1: Given information:

Step 2: Find e^(At):

Step 3: calculation:

Solution

Step by step

Solved in 4 steps with 4 images

SEE SOLUTION Check out a sample Q&A here

Similar questions

An n x n matrix A is skew symmetric if A^(-1)=A^t. Prove that if A is invertible and skew-symmetric that A^(-1) is also skew symmetric (use the properties of inverses.) You need to show that (A^(-1))^t=-A^(-1) . Select the antecedent of the theorem you are proving, then prove the theorem in the box provided. If attaching a file, use only pdf format.
(b) Compute A^2. What is the meaning of the number in the first row and first column of A2? (c) Could the matrix A^2 be the adjacency matrix of a simple graph? If so, draw the graph. If not, explain why.
2. Find a CR-decomposition for each complex matrix A and state the rank of A. -1 1 2+ i ^=671²1 i (a) A = 3 (b) A = 10i 6 - 2i -15i -9 -6+3i
The matrix that rotates by 3π 4 is X' -√2-√2 1 - 1/2 √2
Calculate the matrix A for which (matrix in the attached image), please if able give explanation with the taken steps and show the motivation behind the steps, I am very new to differential equations, thank you in advance.