we get the system:
We know that these equations hold when , because the Cayley-Hamilton theorem
tells us that ! So we get a different system with
[p^2 + qr; pq + qs] + (a + d) [p q] + (ad - bc)[1 0] = [0 0]
[rp + sr; rq + s^2] [r s] [0 1] [0 0]
p = a, q = b, r = c, s = d,
still with four equations, that we know is equal to zero! This means we have four
a, b, c, d and four equations, and we know that these equations are true
for all . But if a polynomial vanishes on infinitely many points, it must
identically be zero. Thus, this means that
ch(A) is the zero polynomial, or
ch(A) = 0
R. This seems to depend on the fact that the ring is infinite, because otherwise
imagine we send to . Since we don't have an infinite number
of elements, why should the polynomial be zero? I imagine that this
needs zariski like arguments to be handled.
§ Cramer's rules
We can get cramer's rule using some handwavy manipulation
or rigorizing the manipulation using geometric algebra.
Say we have a system of equations:
We can write this as:
where and so on. To solve the system, we wedge with and :
Which is exactly cramer's rule.
§ The formula for the adjugate matrix from Cramer's rule (TODO)