Find all matrix X such that $X^2 = \left[ {\begin{array}{} 1 & 0 & 1\\ 0 & 4 & 2\\ 0 & 0 & 16 \end{array}} \right]$

Question

Find all matrix $X$ such that $$X^2 = \begin{bmatrix} 1 & 0 & 1\\ 0 & 4 & 2\\ 0 & 0 & 16 \end{bmatrix}$$

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I think the eigenvalues of $X^2$ are $1, 4, 16$ so $X$ has eigenvalues $1, 2, 4$. Could you please tell me what can I do next?

Answer 1

By a direct calculation there are eight solutions over a field of characteristic zero. They are given by $$ X=\pm \begin{pmatrix} 1 & 0 & \frac{1}{5} \cr 0 & 2 & \frac{1}{3} \cr 0 & 0 & 4\end{pmatrix},\;\pm \begin{pmatrix} -1 & 0 & -\frac{1}{5} \cr 0 & 2 & -1 \cr 0 & 0 & -4\end{pmatrix}, \ldots $$ where the eight different matrices arise by the signs of the eigenvalues, i.e., $\pm (1,2,4),\pm (-1,2,-4),\pm (-1,-2,4),\pm (1,-2,-4)$.