Cryptography is a field of computer science and mathematics that focusses on techniques for secure communication between two parties while a third-party is present. It includes an encryption and decryption. In this paper we study using the multiplicative inverse of the matrix as an application of linear algebra over Z_n in cryptography. We investigat using it in cryptography as the involutory key based on a general matrix formula, and as private key based on a square non-singular matrix. In both cases, we work on the general ring of integers Z_26; as we investigate using it in the last two cases depending on American Standard Code for Information Interchange (ASCII), in this case we work on Z_255. We give algorithms for all that and explain them by examples. Using these keys in cryptography we get secure and confidential messages and communications.