  We present an explicit solution of the eigen-spectrum Toeplitz matrix
$C_{ij}= e^{- \kappa |i-j|}$ with $0\leq i,j \leq N$ and apply it to find
analytically the plasma modes of a layered assembly of 2-dimensional electron
gas. The solution is found by elementary means that bypass the Wiener-Hopf
technique usually used for this class of problems. It rests on the observation
that the inverse of $C_{ij}$ is effectively a nearest neighbor hopping model
with a specific onsite energies which can in turn be diagonalized easily.
Extensions to a combination of a Toeplitz and Hankel matrix, and to a
generalization of $C_{ij}$, are discussed at the end of the paper.
