On some combinatorial identities related to super Catalan matrix

In this paper, we define a new matrix $ S_{n} $ constructed by super Catalan numbers. Also, we give Cholesky and LU-decompositions, Hermite normal form, and the determinant of the matrix $ S_{n} $. Moreover, we derive auxiliary results involving some summation formulas via the coefficients of Lucas polynomials and scaled coefficients of Chebyshev polynomials. Additionally, we give a matrix $ \acute{S}_{n} $ by modifying the matrix $ S_{n} $ to deduce a matrix identity related to matrices $ S_{n} $ and $ \acute{S}_{n}. $ By using the decomposition method, we give an application of solving a system of linear equations of order $ n $ with coefficients $ S(m, n) $ and find a general solution.