Natural matrices, introduced by the first author more than twenty years ago, are very important in several areas such as computational security, cryptography, dynamical systems, differential equations, biology and physics, among others researches that are currently in progress. The main aim of this paper is to present new algebraic results in natural matrices which were included in the first part of the master thesis in applied mathematics of the first author, supervised by the second author. These results are original and correspond to the master thesis of the first author, supervised by the second author. We rewrite some classical results of abstract and linear algebra through Natural Matrices. Finally, we establish beautiful and useful relationships between Natural Matrices and Magic Squares, which are very important in recreational mathematics