We consider the following two problems: to construct a real symmetric arrow matrix A and to construct a real symmetric tridiagonal matrix A, from a special kind of spectral information: one eigenvalue lambda((j)) of the (j x j) leading principal submatrix A (j) of A, j = 1, 2,..., n; and one eigenpair (lambda((n)), x) of A. Here we give a solution to the first problem, introduced in [J. Peng, X.Y. Hu, and L. Zhang. Two inverse eigenvalue problems for a special kind of matrices. Linear Algebra Appl., 416:336-347, 2006.]. In particular, for both problems to have a solution, we give a necessary and sufficient condition in the first case, and a sufficient condition in the second one. In both cases, we also give sufficient conditions in order that the constructed matrices be nonnegative. Our results are constructive and they generate algorithmic procedures to construct such matrices.