<s>[INST] Is there a matrix that is NOT I, but also has eigenvalue of 1? I am looking for a matrix that is not I, but also has eigenvalue of 1. Are there any? Can someone please show me an example and how it has eigenvalue 1.
 [/INST] Here is a simple example:  set
$D = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}; \tag 1$
we have
$D \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \end{pmatrix}, \tag 2$
and
$D \begin{pmatrix} -1 \\ 1 \end{pmatrix} =  \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \begin{pmatrix} -1 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \\ -1 \end{pmatrix} = -\begin{pmatrix} -1 \\ 1 \end{pmatrix}; \tag 3$
we see from (2) and (3) that the eigenvalues of $D$ are $\pm 1$.  We can also find the eigenvalues of $D$ by evaluating the characteristic polynomial $p_D(\mu)$:
$p_D(\mu) = \det(D - \mu I) = \det \left ( \begin{bmatrix} -\mu & 1 \\ 1 & -\mu \end{bmatrix} \right ) = \mu^2 -1; \tag 4$
the roots of
$p_D(\mu) = \mu^2 - 1 \tag 5$
are $\mu = \pm 1$, in agreement with (2)-(3).
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