Given matrix $A$, find the number of solutions of $X^2 = A$.

Question

Given the matrix:

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

Find the number of solutions (in $M_3(\mathbb{R})$) of the equation $X^2 = A$.

I tried writing $X$ as:

$$ X = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{pmatrix} $$

And then I multiplied $X$ by itself and created a system with the terms of $X^2 = A$, but $X^2$ is very messy and I got stuck. I couldn't solve this problem. Is there another (and possibly better) approach to this?

Answer 1

The determinant of $A$ can be seen to be $-1$, so if $X^2=A$, then $\det(X)^2=\det(A)=-1$, which is impossible.