Are there uncountably many $A\in M_3 (\mathbb {R})$ such that $A^8=I $?

linear-algebra matrices matrix-equations minimal-polynomials

Hint: What about $$\begin{bmatrix}1&-t&0\\0&1&0\\0&0&1\end{bmatrix}\begin{bmatrix}1&0&0\\0&-1&0\\0&0&-1\end{bmatrix} \begin{bmatrix}1&t&0\\0&1&0\\0&0&1\end{bmatrix}$$


Hint The reflection $R$ about any $2$-dimensional subspace satisfies $R^2 = I$.

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