  A method of generating differential operators is used to solve the spectral
problem for a generalisation of the Sylvester-Kac matrix. As a by-product, we
find a linear differential operator with polynomial coefficients of the first
order that has a finite sequence of polynomial eigenfunctions generalising the
operator considered by M. Kac. In addition, we explain spectral properties of
two related tridiagonal matrices whose shape differ from our generalisation.
