Elements of order $7$ and rational canonical forms.

Elements of order $7$ and rational canonical forms.

The prime factorization of $x^7-1$ in $\Bbb{Z}_2[x]$ is
$$x^7-1=(x-1)(x^3+x^2+1)(x^3+x+1).$$
Recall that for any field $F$, a matrix $A \in Gl_n(F)$ satisfies $A^7=I$
(i.e., $A$ has order dividing $7$) if and only if the minimal polynomial
of $A$ divides $x^7-1$.
a) Give the rational canonical forms of all elements of order exactly $7$
in $Gl_3(\Bbb{Z}_2)$. (Write down the specific $3 \times 3$ matrices).
b) Give the rational canonical forms of all elements of order exactly $7$
in $Gl_6(\Bbb{Z}_2)$. Here you may use $C(f)$ notation and specify
appropriate block sums for appropriate $f$'s.
a) Since any matrix satisfying $x-1$ has order 1, the other two are
possible minimal polynomials for matrices of order $7$. But if a matrix
satisfies $x^3+x^2+1$ or $x^3+x+1$, then it must be of order either $1$ or
$7$. Since it can't be $1$, we have
$$\begin{bmatrix} 0 & 0 & -1 \\1 & 0 & 0 \\0 & 1 & -1 \end{bmatrix}$$
and
$$\begin{bmatrix} 0 & 0 & 0 \\1 & 0 & -1 \\0 & 1 & -1 \end{bmatrix}$$
as the possible rational canonical forms.
b) We have
$$C(x^3+x^2+1) \oplus C(x^3+x^2+1)$$
$$C(x^3+x+1) \oplus C(x^3+x+1)$$
$$C(x^3+x+1) \oplus C(x^3+x^2+1)$$
Is that correct?
Thanks in advance