Let A be an eigenvalue of an invertible matrix A. Show that A-1 is an eigenvalue of A¯!. [Hint: Suppose a nonzero x satisfies Ax = Ax.] Find the characteristic polynomial and the real eigenvalues of the matrices below. (b) [ % 3 (a) 6. -4 4
Let A be an eigenvalue of an invertible matrix A. Show that A-1 is an eigenvalue of A¯!. [Hint: Suppose a nonzero x satisfies Ax = Ax.] Find the characteristic polynomial and the real eigenvalues of the matrices below. (b) [ % 3 (a) 6. -4 4
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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![### Eigenvalue Problem
**Theorem:**
Let \( \lambda \) be an eigenvalue of an invertible matrix \( A \). Show that \( \lambda^{-1} \) is an eigenvalue of \( A^{-1} \).
*Hint: Suppose a nonzero \( \mathbf{x} \) satisfies \( A\mathbf{x} = \lambda\mathbf{x} \).*
---
**Problem:**
Find the characteristic polynomial and the real eigenvalues of the matrices below.
(a)
\[
\begin{bmatrix}
-4 & -1 \\
6 & 1
\end{bmatrix}
\]
(b)
\[
\begin{bmatrix}
5 & 3 \\
-4 & 4
\end{bmatrix}
\]
**Task:** Analyze each matrix to determine the characteristic polynomial and calculate the real eigenvalues.
---
This exercise involves finding eigenvalues, which are crucial for understanding matrix properties in linear algebra and many applied fields such as physics and engineering.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F07fffa04-d151-44be-8465-53f07054b4bf%2F8e211851-1f17-4a56-a34b-6f6fde1f6fa0%2Fl0etel_processed.png&w=3840&q=75)
Transcribed Image Text:### Eigenvalue Problem
**Theorem:**
Let \( \lambda \) be an eigenvalue of an invertible matrix \( A \). Show that \( \lambda^{-1} \) is an eigenvalue of \( A^{-1} \).
*Hint: Suppose a nonzero \( \mathbf{x} \) satisfies \( A\mathbf{x} = \lambda\mathbf{x} \).*
---
**Problem:**
Find the characteristic polynomial and the real eigenvalues of the matrices below.
(a)
\[
\begin{bmatrix}
-4 & -1 \\
6 & 1
\end{bmatrix}
\]
(b)
\[
\begin{bmatrix}
5 & 3 \\
-4 & 4
\end{bmatrix}
\]
**Task:** Analyze each matrix to determine the characteristic polynomial and calculate the real eigenvalues.
---
This exercise involves finding eigenvalues, which are crucial for understanding matrix properties in linear algebra and many applied fields such as physics and engineering.
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