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.