In this study, firstly, we will present some properties of hyperbolic numbers. Then, we will introduce hyperbolic matrices, which are matrices with hyperbolic number entries. Additionally, we will examine the algebraic properties of these matrices and reveal its difference from other matrix structures such as real, dual, and complex matrices. As a result of comparing the results found in this work with real, dual, and complex matrices, it will be revealed that there are similarities in additive properties and differences in some multiplicative properties. Finally, we will define some special hyperbolic matrices and give their properties and relations with real matrices.