Once upon a less enlightened time, when people were less knowledgeable in the intricacies of algorithmically computing eigenvalues, methods for generating the coefficients of a matrix's eigenpolynomial were quite widespread. One of the more prominent methods for computing the coefficients was a method ascribed to both the Frenchman Leverrier, and the Russian Faddeev (who was an (co-)author of one of the oldest references on the practice of numerical linear algebra).
The (Faddeev-)Leverrier method is a method that will require you to do a number of matrix multiplications to generate the coefficients of the characteristic polynomial. Letting the $n\times n$ matrix $\mathbf A$ have the monic characteristic polynomial $(-1)^n \det(\mathbf A-\lambda\mathbf I)=\lambda^n+c_{n-1}\lambda^{n-1}+\cdots+c_0$, the algorithm proceeds like so:
$\mathbf C=\mathbf A;$
$\text{for }k=1,\dots,n$
$\text{if }k>1$
$\qquad \mathbf C=\mathbf A\cdot(\mathbf C+c_{n-k+1}\mathbf I);$
$c_{n-k}=-\dfrac{\mathrm{tr}(\mathbf C)}{k};$
$\text{end for}$
If your computing environment can multiply matrices, or take their trace (sum of the diagonal elements, $\mathrm{tr}(\cdot)$), then you can easily program (Faddeev-)Leverrier. The method works nicely in exact arithmetic, or in hand calculation (assuming you have the stamina to repeatedly multiply matrices), but is piss-poor in inexact arithmetic, as the method tends to greatly magnify rounding errors in the matrix, ever yielding coefficients that become increasingly inaccurate as the iteration proceeds. But, for the simple $3\times 3$ case envisioned by the OP, this should work nicely.
People interested in this old, retired method might want to see this paper.
By the Cayley-Hamilton theorem, we have $(A-1)(A+2)^2(A-3)=0$, that is,
$A^4-9A^2-4A+12I=0$.
Multiply both sides by $A^{-1}$, and be amazed!
Best Answer
Let $M$ be a matroid on ground set $E$, we have $$\rho_M(\lambda) = \sum_{S \subseteq E}(-1)^{|S|}\lambda^{r(M) - r(S)}$$ where notice we use the corank as our exponent instead of the rank. If our matroid has no loops then $r(\varnothing) = 0$ and $\varnothing$ is the only set of rank zero and we see that $\rho_M$ is monic with degree $r(M)$. (This property is shared by the characteristic polynomial of a graded poset; see the Mobius function definite of $\rho_M$ in the Wikipedia article you have linked to). We can go further in the case of no loop every singleton subset has rank one so the coefficient of $\lambda^{r(M)-1}$ is $-|E|$. Next notice any subset of size two has rank two unless we have a two element circuit. Simple matroids have no one or two element circuits. So, for a simple matroid the coefficient of $\lambda^{r(M)-2}$ is $\binom{|E|}{2}$. More generally the coefficients have an interpretation in terms of "no broken circuit" (NBC) sets. Whitney has a theorem about the coefficients of the chromatic polynomial of a graph (which is a special case of the characteristic polynomial of a matroid up to a factor of some power of $\lambda$) in terms of NBC sets. Also Rota had an NBC theorem of matroids/geometric lattices (here note geometric lattices are exactly lattices of flats for matroids).
Also in the Wikipedia article you linked to it discusses the the Tutte polynomial of a matroid. Notice the characteristic polynomial is just an evaluation of the Tutte polynomial. That is the Tutte polynomials in a more general two variable polynomial with lot of nice properties. Let $T_M(x,y)$ be the Tutte polynomial of a matroid. Then $T_{M^*}(x,y) = T_M(y,x)$ thus duality in the Tutte polynomial is just exchanging variable. Also $T_{G}(x,y) = T_{M_G}(x,y)$ so again matroid polynomial generalizes the graph polynomial. The Tutte polynomial behaves well with respect duality as we already saw and also there is a deletion contraction rule. Finally the Tutte polynomial can be used to count things. For example $T_G(1,1)$ is the number of spanning forest in a graph and $T_M(1,1)$ is the number of bases of your matroid. See Theorem 1.8 of this paper for more evaluations of the Tutte polynomial.
So, to directly answer some of your question. I do not know what specific evaluations $\rho_M(\lambda)$ are in the non-graphic case. I think the main motivation for the matroid polynomials are to generalize what we know for graphs.