Is an Eigenvector of a Matrix an Eigenvector of its Inverse?
Problem 70
Suppose that $A$ is an $n \times n$ matrix with eigenvalue $\lambda$ and corresponding eigenvector $\mathbf{v}$.
(a) If $A$ is invertible, is $\mathbf{v}$ an eigenvector of $A^{-1}$? If so, what is the corresponding eigenvalue? If not, explain why not.
(b) Is $3\mathbf{v}$ an eigenvector of $A$? If so, what is the corresponding eigenvalue? If not, explain why not.
(Stanford University, Linear Algebra Exam)
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