Fundamental matrix operations and linear systems of equations are crucial for development of varieties of scientific fields. In this paper we propose an alternative quantum protocol for some matrix operations including matrix-vector product, matrix-matrix product, sum of two matrices, calculation of determinant and inverse matrix. With the "Sender-Receiver" model, we encode the matrix entries into the probability amplitudes of pure initial states of senders. After applying proper unitary transformation to the complete quantum system, the result can be found in the certain blocks of the receiver's density matrix. Furthermore, based on the mentioned quantum algorithms we present a new quantum algorithm for solving linear systems of equations. These quantum protocols can be used as subroutines in other quantum schemes.