Calculation of the $3\times 3$ exponent matrix via Cayley-Hamilton theorem.

Question

I have a random $3\times 3$ matrix $A$.

How can I calculate $e^A$ by $E$ (the identity matrix), $A$ and $A^2$, using the Cayley-Hamilton theorem?

I need a general expression that includes only the trace, determinant and signature (tensor invariants) of a given matrix $A$.