$A=\begin{pmatrix}0&0&1\\ 0&1&0\\ 1&0&0\end{pmatrix}$, $X^2=A$ has no solutions

Question

$A=\begin{pmatrix}0&0&1\\ 0&1&0\\ 1&0&0\end{pmatrix}$

How do I show that there are no solutions for $X^2=A$ in $M_3(R)$?

I tried $X^2=\begin{pmatrix}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\end{pmatrix}^2=A$, but I couldn't seem to find anything that shows there are no solutions. How should I approach this?

Answer 1

Hint: $X^2=A$ implies $(\det X)^2 = \det A$. What is $\det A$?

Answer 2

Hint:

Determine $\det A$ and then equate it to $\det X^{2}$.