For an $n\times n$ matrix $A$, using the Cayley-Hamilton theorem, we can express
$$e^{tA} = \alpha_1(t)I + \alpha_2(t)A + \cdots + \alpha_n(t)A^{n-1}$$
where the coefficients $\alpha_i$ could easily be found by replacing the matrix by eigenvalues in the above expression or its derivatives in case of repeated eigenvalues.
Why do we need to find eigenvectors at all to calculate the matrix exponential, or even to find solutions to the systems of linear differential equations, when we can find solutions without eigenvectors?