Companion matrices over the Galois field with two elements

Question

Let us denote $\mathbb{F}_2$ by the Galois field with two elements. The companion matrix of the monic polynomial $f(x)=a_0+a_1x+a_2x^2+\cdots+a_{n-1}x^{n-1}+x^n$ is the square matrix defined as $$C(f)=\begin{pmatrix} 0& & &-a_0 \\ 1& & &-a_1\\ &\ddots& &\vdots\\ 0& &1&-a_{n-1} \end{pmatrix}.$$ I am considering about the companion matrix $A$ of $x^n+1$ over $\mathbb{F}_2$. I wonder if there exists a matrix $B$ such that $A=B^2$. I tried it but got no results. If not, can it express into commutators of involutions? I know for sure that it is a product of two involutions.