9 Tr(-2A) = -6, det(-2A) 28.
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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I'm currently struggling with solving this problem using matrix notation alone, and I'm seeking your assistance. The requirement is to find a solution using matrix notation exclusively, without incorporating any other approaches. Could you kindly provide a detailed, step-by-step explanation in matrix notation to guide me towards the final solution?
![Here is the transcription of the text for an educational website:
---
### Problem 9.19
Given the matrix \( A \), the trace and determinant of the matrix \(-2A\) (where \(-2A\) is obtained by multiplying each element of \(A\) by -2) are provided as follows:
\[
\text{Tr}(-2A) = -6, \quad \det(-2A) = 28.
\]
---](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0113a2ae-503a-4a14-a5d4-d8fff765856f%2F247510ea-dab8-45ca-ac88-dde55d1d3040%2Fsnqqx0t_processed.png&w=3840&q=75)
Transcribed Image Text:Here is the transcription of the text for an educational website:
---
### Problem 9.19
Given the matrix \( A \), the trace and determinant of the matrix \(-2A\) (where \(-2A\) is obtained by multiplying each element of \(A\) by -2) are provided as follows:
\[
\text{Tr}(-2A) = -6, \quad \det(-2A) = 28.
\]
---
![### Matrix Characteristic Polynomial and Transformations
#### Problem
Given the characteristic polynomial of a matrix \( A \) as \( p(\lambda) = \lambda^2 - 3\lambda + 7 \), determine the trace and the determinant of the matrix \( -2A \).
#### Explanation
- **Characteristic Polynomial**: The characteristic polynomial of a matrix \( A \) of size \( n \times n \) is \( p(\lambda) = \mathrm{det}(\lambda I - A) \), where \( I \) is the identity matrix.
- For a \( 2 \times 2 \) matrix \( A \) with a characteristic polynomial \( p(\lambda) \), we can write \( p(\lambda) = \lambda^2 - (\text{Tr}(A))\lambda + \det(A) \), where:
- \( \text{Tr}(A) \) (trace of \( A \)) is the sum of the diagonal elements of \( A \).
- \( \det(A) \) (determinant of \( A \)) is the product of the eigenvalues of \( A \).
Given:
\[ p(\lambda) = \lambda^2 - 3\lambda + 7 \]
From the polynomial, we can identify that:
- Trace of \( A \), \( \text{Tr}(A) = 3 \)
- Determinant of \( A \), \( \det(A) = 7 \)
#### Calculations for \( -2A \)
- **Trace**: The trace of \( -2A \) is given by \( -2 \times \text{Tr}(A) \):
\[
\text{Tr}(-2A) = -2 \times 3 = -6
\]
- **Determinant**: The determinant of \( -2A \) for a \( 2 \times 2 \) matrix is given by \( (-2)^2 \times \det(A) \):
\[
\det(-2A) = 4 \times 7 = 28
\]
#### Conclusion
- The trace of \( -2A \) is \(-6\).
- The determinant of \( -2A \) is \( 28 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0113a2ae-503a-4a14-a5d4-d8fff765856f%2F247510ea-dab8-45ca-ac88-dde55d1d3040%2Fha4tev_processed.png&w=3840&q=75)
Transcribed Image Text:### Matrix Characteristic Polynomial and Transformations
#### Problem
Given the characteristic polynomial of a matrix \( A \) as \( p(\lambda) = \lambda^2 - 3\lambda + 7 \), determine the trace and the determinant of the matrix \( -2A \).
#### Explanation
- **Characteristic Polynomial**: The characteristic polynomial of a matrix \( A \) of size \( n \times n \) is \( p(\lambda) = \mathrm{det}(\lambda I - A) \), where \( I \) is the identity matrix.
- For a \( 2 \times 2 \) matrix \( A \) with a characteristic polynomial \( p(\lambda) \), we can write \( p(\lambda) = \lambda^2 - (\text{Tr}(A))\lambda + \det(A) \), where:
- \( \text{Tr}(A) \) (trace of \( A \)) is the sum of the diagonal elements of \( A \).
- \( \det(A) \) (determinant of \( A \)) is the product of the eigenvalues of \( A \).
Given:
\[ p(\lambda) = \lambda^2 - 3\lambda + 7 \]
From the polynomial, we can identify that:
- Trace of \( A \), \( \text{Tr}(A) = 3 \)
- Determinant of \( A \), \( \det(A) = 7 \)
#### Calculations for \( -2A \)
- **Trace**: The trace of \( -2A \) is given by \( -2 \times \text{Tr}(A) \):
\[
\text{Tr}(-2A) = -2 \times 3 = -6
\]
- **Determinant**: The determinant of \( -2A \) for a \( 2 \times 2 \) matrix is given by \( (-2)^2 \times \det(A) \):
\[
\det(-2A) = 4 \times 7 = 28
\]
#### Conclusion
- The trace of \( -2A \) is \(-6\).
- The determinant of \( -2A \) is \( 28 \).
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