Hints:

- one can easily convert a list of numbers into a polynomial.
- one can easily convert a polynomial into a finite field element.

Here is an example which hopefully addresses the question.

Define a number field:

```
sage: p = 29
sage: n = 2
sage: F.<x> = GF(p^n)
```

Define a polynomial ring over the integers.

```
sage: R = PolynomialRing(ZZ, 't')
```

Now, convert a list of coefficients (from degree 0 to degree n) into a polynomial.

```
sage: u = [7, 4] # careful: start from degree 0 coefficient
sage: g = R(u)
sage: g
4*t + 7
```

Now, turn this polynomial into a finite field element:

```
sage: z = F(g)
sage: z
4*x + 7
```

Combining this (again, careful with order of coefficients),
define lists of lists of coefficients:

```
sage: u1 = [[7, 4], [1, 3]]
sage: u2 = [[1, 3], [2, 5]]
```

turn them into vectors:

```
sage: v1 = vector(F, [R(u) for u in u1])
sage: v2 = vector(F, [R(u) for u in u2])
```

and now you can add them:

```
sage: v1 + v2
(7*x + 8, 8*x + 3)
```

Please describe mathematically what is needed, then we can see how to come together. Simple questions: We want a $2\times 2$ matrix with entries in the polynomial ring over some finite field? Please use then LaTeX to declare it explicitly. Please define $n$, better, give an example with a fixed $n$.

Is this a correct rephrasing of your question?