  We have calculated the joint probability distribution function for random
reverse-cyclic matrices and shown that it is related to an N-body exactly
solvable model. We refer to this well-known model potential as a screened
harmonic oscillator. The connection enables us to obtain all the correlations
among the particle positions moving in a screened harmonic potential. The
density of nontrivial eigenvalues of this ensemble is found to be of the Wigner
form and admits a hole at the origin, in contrast to the semicircle law of the
Gaussian orthogonal ensemble of random matrices. The spacing distributions
assume different forms ranging from Gaussian-like to Wigner.
