Why is this matrix not a reflection in a plane matrix?

Question

$$ \begin{pmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \\ \end{pmatrix} $$

This matrix clearly has det -1 and is orthogonal. Why is it not reflection in a plane matrix?

Answer 1

It is easy to see that this matrix corresponds to the reflection through the origin $(0,0,0)$ hence the origin is the only invariant point of this tranformation.

Answer 2

In your definition of a reflection of $\mathbb{R}^3$, you have to replace your condition $\det = -1$ by a condition about the dimensions of the eigenspaces $E_1$ and $E_{-1}$ associated to the eigenvalues $1$ and $-1$, namely that $\dim E_1=2$ and $\dim E_{-1}=1$ (note that this condition automatically implies that your determinant is $-1$). Here we clearly have that $\dim E_{-1}=3$.