  We present here the necessary and sufficient conditions for the invertibility
of tridiagonal matrices, commonly named Jacobi matrices, and explicitly compute
their inverse. The techniques we use are related with the solution of
Sturm-Liouville boundary value problems associated to second order linear
difference equations. These boundary value problems can be expressed throughout
a discrete Schr\"odinger operator and their solutions can be computed using
recent advances in the study of linear difference equations. The conditions
that ensure the uniqueness solution of the boundary value problem lead us to
the invertibility conditions for the matrix, whereas the solutions of the
boundary value problems provides the entries of the inverse matrix.
