Some New Properties of Frank Matrices with Entries Mersenne Numbers

In this work, we give some new properties of Frank matrices \(F[a_n]=[a_{ij}]_{i,j=1}^n\) $F[a_n]={[a_{\mathit{ij}}]}_{i,j=1}^n$ such that \(a_{ij}=M_{n+1-max\{i,j\}}\) $a_{\mathit{ij}}=M_{n+1-max\{i,j\}}$ or \(R_{n+1-max\{i,j\}}\) $R_{n+1-max\{i,j\}}$ , where \(M_n\) $M_n$ ( \(R_n\) $R_n$ ) is the nth Mersenne (Fermat) numbers. Here, we obtain some algebraic properties of these matrices such as determinants, matrix norms, characteristic polynomials, spread, etc. In support of the results, we provide numerical validations.