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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
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:** - **
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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