I am taking an abstract algebra course and am getting quite confused with the terminology of invariant factors, elementary divisors, and the normal forms. I am asked to compute the rational canonical form of a matrix $A$ whose entries are all $1 \in \mathbb F_p$, for some prime $p$.
I am not asking for an answer, I would much rather be pointed in the correct direction with hints or resources. Here is what I currently believe to understand:
I'm very sure the minimal polynomial of this matrix is $m(x) = x - 1$, since the matrix has all entries of $1$. I thought the minimal polynomial was used in the calculation of the rational canonical form, but am failing to find a resource on how $m(x)$ can be of any help.
I also attempted to compute the Smith normal form of an example $2 \times 2$ matrix $A - xI$ to find it's invariant factors, and arrived at
$$ SNF(A - xI) = \left( \begin{array}{c c} 1 & 0\\ 0 & x(x - 2) \end{array} \right)$$
From that, I believe the invariant factor is merely $x(x - 2)$ and thus the rational canonical form is $$ RCF(A) = \left( \begin{array}{c c} 0 & 0\\ 1 & 2 \end{array} \right)$$
So, my main questions are, is this correct? And, since the entries are in $\mathbb F_p$, how should I go about handling $n\times n$ matrices where $n \ge p$.