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QUESTION

Show that if $\lambda$ is a non-zero eigenvalue of the invertible
$n\times n$ matrix $A$ then $\lambda^{-1}$ is an eigenvalue of
$A^{-1}$. Illustrate the theorem with a matrix of your choice.

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ANSWER

If $A\textbf{x}=\lambda\textbf{x}$ then
$\frac{1}{\lambda}\textbf{x}=A^{-1}\textbf{x}$. Any invertible
matrix will do for the illustration (but one hopes that students
will not choose a $1\times 1$ matrix).



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