A Universe of Sorts

I want to 'implement' the zariski based proof for cayley hamilton in SAGE and show that it works by checking the computations scheme-theoretically.
Let's work through the proof by hand. Take a 2x2 matrix [a, b; c, d].
The charpoly is |[a-l; b; c; d-l]| = 0, which is p(l) = (a-l)(d-l) - bc = 0
This simplified is p(l) = l^2 - (a + d) l + ad - bc = 0.
Now, let's plug in l = [a; b; c; d] to get the matrix eqn
[a;b;c;d]^2 - (a + d)[a;b;c;d] + [ad - bc; 0; 0; ad - bc] = 0.
The square is going to be [a^2 +]
Let X be the set of (a, b, c, d) such that the matrices [a;b;c;d] satisfy their only charpoly.
Consider the subset U of the set (a, b, c, d) such that the matrix [a;b;c;d] has distinct eigenvalues.
For any matrix with distinct eigenvalues, it is easy to show that they satisfy their charpoly.
First see that diagonal matrices satisfy their charpoly by direct computation: [a;0;0;b] has eigenvalues (a, b). Charpoly is l^2 - l(a + b) + ab. Plugging in the matrix, we get [a^2;0;0;b^2] - [a(a+b);0;0;b(a+b)] + [ab;0;0;ab] which cancels out to 0.
Then note that similar matrices have equal charpoly, so start with |(λI - VAV')| = 0. rewrite as (VλIV' - VAV') = 0, which is V(λI - A)V' = 0, which is the same λI - A = 0.
Thus, this means that a matrix with distinct eigenvalues, which is similar to a diagonal matrix (by change of basis), has a charpoly that satisfies cayley hamilton.
Thus, the set of matrices with distinct eigenvalues, U is a subset of X.

However, it is not sufficient to show that the system of equations has an infinite set of solutions.
For example, xy = 0 has infinite solutions (x=0, y=k) and (x=l, y=0), but that does not mean that it is identically zero.
This is in stark contrast to the 1D case, where a polynomial p(x) = 0 having infinite zeroes means that it must be the zero polynomial.
Thus, we are forced to look deeper into the structure of solution sets of polynomials, and we need to come up with the notion of irreducibility.
See that the space K^4 is irreducible, where K is the field from which we draw coefficients for our matrix.

Next, we note that X is a closed subset of k^4 since it's defined by the zero set of the polynomial equations.
We note that U is an open subset of k^4 since it's defined as the non-zero set of the discriminant of the charpoly! (ie, we want non-repeated roots)
Also note that U is trivially non-empty, since it has eg. all the diagonal matrices with distinct eigenvalues.
So we have a closed subset X of k^4, with a non-empty open subset U inside it.
But now, note that the closure of U must lie in X, since X is a closed set, and the closure U of the subset of a closed set must lie in X.
Then see that since the space is irreducible, the closure of U (an open) must be the whole space.
This means that all matrices satisfy cayley hamilton!

§ Cayley Hamilton for 2x2 Matrices in Sage via AG