Similar matrices over $\mathbb{Z}/2\mathbb{Z}$

Question

Given the following matrices $P=\left( \begin{array}{rrr} 1 & -1 & 0 \\ 0 & 2 & 5 \\ 0 & 0 & 3 \\ \end{array}\right), Q=\left( \begin{array}{rrr} 1 & 0 & 0 \\ -1 & 4 & 0 \\ 0 & 3 & 7 \\ \end{array}\right)$,
such that $P,Q \in M(3\times3, \mathbb{Z}/2\mathbb{Z})$, I have to find out whether the two matrices are similar or not.

One problem I'm facing is that $\mathbb{Z}/2\mathbb{Z} = {0,1}$, which implies that $-1,2,5,3,4,7 \notin \mathbb{Z}/2\mathbb{Z}$, hence I'm assuming it is meant that

$P=\left( \begin{array}{rrr} 1 & (-1\mod2) & 0 \\ 0 & (2\mod2) & (5\mod2) \\ 0 & 0 & (3\mod2) \\ \end{array}\right)\\ Q=\left( \begin{array}{rrr} 1 & 0 & 0 \\ (-1\mod2) & (4\mod2) & 0 \\ 0 & (3\mod2) & (7\mod2) \\ \end{array}\right)$

which would give us

$P=\left( \begin{array}{rrr} 1 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 1 \\ \end{array}\right)\\ Q=\left( \begin{array}{rrr} 1 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 1 \\ \end{array}\right)$

But now the real problem arises, which is finding $R\in M(3\times3, \mathbb{Z}/2\mathbb{Z})$, such that $P=RQR^{-1}$.

One attempt was to try and solve a huge system of equations, arising ,for example, from $P=RQR^{-1}$ but that does not seem like the proper way to do this (it didn't work out).
Another thing I noticed is that $P, Q$ have the same rank, however, I was unable to use this in order to find a solution.
My last resort so far would be to simply try out various matrices in $M(3\times3, \mathbb{Z}/2\mathbb{Z})$, but this can't be the only way to do this.

Any help is appreciated, thanks.

Answer 1

Just by looking at the matrices, it seems that $$R=\pmatrix{0&0&1\\0&1&0\\1&0&0}=R^{-1}$$ should work. It permutes the first and last columns, and then the first and last rows.

Answer 2

Yes, they are similar. They have the same characteristic polynomial, which is $x^3+x=x(x+1)^2$. So, each of them is similar to$$\begin{bmatrix}0&0&0\\0&1&0\\0&0&1\end{bmatrix}\text{ or to }\begin{bmatrix}0&0&0\\0&1&1\\0&0&1\end{bmatrix}.$$But you can easily check that, both for $P$ and for $Q$, the eigenspace associated with the eignvalue $1$ is $1-$ dimensional. Therefore, they are both similar to the second of the two matrices mentioned above, and so they are similar to each other.