The exponential matrix: an explicit formula by an elementary method

We show an explicit formula, with a quite easy deduction, for the exponential matrix $e^{tA}$ of real and finite square $A$ (and complex ones also). The elementary method developed avoids Jordan canonical form, eigenvectors, resolution any linear system, inversion, polynomial interpolation, integration, functional analysis, generalized Fibonacci sequences. basic tools are Cayley-Hamilton theorem partial fraction decomposition. Two examples given. also that such applies to algebraic operators on infinite dimensional Banach spaces.