We develop a compact solution formula for a large class of exact solutions (including
the so-called multi-soliton solutions) to the nonlinear partial differential-difference equation
known as the Toda lattice equation. In this model, the nearest neighboring particles interact
with forces that exponentially decrease as a function of the distance between them. In
general, nonlinear equations do not have exact solutions and we can only obtain numerical
solutions for them. However, the Toda lattice equation belongs to a special class known as the
integrable evolution equations and it has certain exact solutions. Our solution formula yields
exact solutions constructed by using three constant matrices. We investigate the relationship
between the mathematical parameters in our matrix triplet and the physical properties of the
waves represented by our solutions. In particular, we relate such mathematical parameters
to the velocities and initial locations of the individual solitons in our solutions. Using the
concept of a Fredholm determinant, we further generalize our solutions to the case where
the matrix size becomes infinite in our matrix triplet.