Let 2 be an eigenvalue of an invertible matrix A. Show that A-1 is an eigenvalue of A -1 [Hint: Suppose a nonzero satisfies Ax = Ax.] ..... -1 exists. In order for A' to be an eigenvalue of A', there must exist a nonzero x such that A 1x =-1x. -1 Note that A Suppose a nonzero satisfies Ax = Ax. What is the first operation that should be performed on Ax = x so that an equation similar to the one in the previous step can be obtained? O A. Invert the product on each side of the equation. O B. Left-multiply both sides of Ax = x by A1. OC. Right-multiply both sides of Ax = Ax by A1. Perform the operation and simplify. |(Type an equation. Simplify your answer.)
Let 2 be an eigenvalue of an invertible matrix A. Show that A-1 is an eigenvalue of A -1 [Hint: Suppose a nonzero satisfies Ax = Ax.] ..... -1 exists. In order for A' to be an eigenvalue of A', there must exist a nonzero x such that A 1x =-1x. -1 Note that A Suppose a nonzero satisfies Ax = Ax. What is the first operation that should be performed on Ax = x so that an equation similar to the one in the previous step can be obtained? O A. Invert the product on each side of the equation. O B. Left-multiply both sides of Ax = x by A1. OC. Right-multiply both sides of Ax = Ax by A1. Perform the operation and simplify. |(Type an equation. Simplify your answer.)
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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![# Educational Content: Eigenvalue and Invertible Matrices
## Problem Statement
Let \(\lambda\) be an eigenvalue of an invertible matrix \(A\). Show that \(\lambda^{-1}\) is an eigenvalue of \(A^{-1}\).
**Hint:** Suppose a nonzero \(x\) satisfies \(Ax = \lambda x\).
## Steps to Solve
1. **Understanding \(A^{-1}\) and Eigenvalues:**
- Given that \(A^{-1}\) exists, for \(\lambda^{-1}\) to be an eigenvalue of \(A^{-1}\), there must be a nonzero \(x\) such that:
\[
A^{-1}x = \lambda^{-1}x
\]
2. **Determine the First Operation:**
- Suppose a nonzero \(x\) satisfies the equation \(Ax = \lambda x\).
- **Question:** What operation should be done on \(Ax = \lambda x\) to obtain an equation similar to the one mentioned above?
- **Options:**
- A. Invert the product on each side of the equation.
- B. Left-multiply both sides of \(Ax = \lambda x\) by \(A^{-1}\).
- C. Right-multiply both sides of \(Ax = \lambda x\) by \(A^{-1}\).
3. **Perform and Simplify the Operation:**
- Choose an option and perform the specified operation.
4. **Justification for \(\lambda^{-1}\):**
- **Question:** Why does this show that \(\lambda^{-1}\) is defined?
- **Options:**
- A. Since \(x\) is an eigenvector of \(A\), \(A^{-1}\) and \(x\) are commutable. Hence, the previous equation is uninterpretable if \(\lambda = \).
- B. By definition, \(x\) is nonzero and \(A\) is invertible. Therefore, the previous equation is unsatisfiable if \(\lambda = 0\).
- C. As \(\lambda^{-1}x\) must be defined and nonzero, \(\lambda^{-1}\) must exist and be nonzero.
5. **Proving \(\lambda^{-1}\) as an Eigenvalue:**
- **](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F75f96cc1-f20f-41e8-af87-e67129ee7f4c%2F57a5af87-c17c-4ad2-8102-4ac7abc40bb1%2Frnkmpea_processed.png&w=3840&q=75)
Transcribed Image Text:# Educational Content: Eigenvalue and Invertible Matrices
## Problem Statement
Let \(\lambda\) be an eigenvalue of an invertible matrix \(A\). Show that \(\lambda^{-1}\) is an eigenvalue of \(A^{-1}\).
**Hint:** Suppose a nonzero \(x\) satisfies \(Ax = \lambda x\).
## Steps to Solve
1. **Understanding \(A^{-1}\) and Eigenvalues:**
- Given that \(A^{-1}\) exists, for \(\lambda^{-1}\) to be an eigenvalue of \(A^{-1}\), there must be a nonzero \(x\) such that:
\[
A^{-1}x = \lambda^{-1}x
\]
2. **Determine the First Operation:**
- Suppose a nonzero \(x\) satisfies the equation \(Ax = \lambda x\).
- **Question:** What operation should be done on \(Ax = \lambda x\) to obtain an equation similar to the one mentioned above?
- **Options:**
- A. Invert the product on each side of the equation.
- B. Left-multiply both sides of \(Ax = \lambda x\) by \(A^{-1}\).
- C. Right-multiply both sides of \(Ax = \lambda x\) by \(A^{-1}\).
3. **Perform and Simplify the Operation:**
- Choose an option and perform the specified operation.
4. **Justification for \(\lambda^{-1}\):**
- **Question:** Why does this show that \(\lambda^{-1}\) is defined?
- **Options:**
- A. Since \(x\) is an eigenvector of \(A\), \(A^{-1}\) and \(x\) are commutable. Hence, the previous equation is uninterpretable if \(\lambda = \).
- B. By definition, \(x\) is nonzero and \(A\) is invertible. Therefore, the previous equation is unsatisfiable if \(\lambda = 0\).
- C. As \(\lambda^{-1}x\) must be defined and nonzero, \(\lambda^{-1}\) must exist and be nonzero.
5. **Proving \(\lambda^{-1}\) as an Eigenvalue:**
- **
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