The authors consider the problem of reconstructing two \(n\times n\) Jacobi matrices \(J_{n},\;J_{n}^{\ast }\) and vectors \(X_{1}\), \(Y_{1}\in \mathbb{R}^{k}\) such that for a given \(k\times k\) Jacobi matrix \(J_{k}\) where \( \left( 1\leq k\leq n-1\right) \), real scalars \(S,\; \lambda,\; \mu \) and vectors \(X_{2},\) \(Y_{2}\in \mathbb{R}^{n-k}\) we either have \(\left( \lambda ,X\right) \) and \(\left( \mu ,Y\right) \) (a) are respectively the maximal eigenpairs of \(J_{n}\) and \(J_{n}^{\ast }\) or(b) are respectively the maximal and the minimal eigenpairs of \(J_{n}\) and \(J_{n}^{\ast }.\) The constant \(S=\cos (\theta)\) depends on the angle between \(X\) and \(Y\). The reconstruction is based on the recursive properties of the determinant of a Jacobi matrix. Numerical algorithms are implemented at the end.