Efficiently processing basic linear algebra subroutines is of great importance for a wide range of computational problems. In this paper, we consider techniques to implement matrix functions on a quantum computer. We embed given matrices into 3 times larger Hermitian matrices and assume as input a given set of unitary operators generated by the embedding matrices. With the matrix embedding formula, we give Trotter-based quantum subroutines for elementary matrix operations include addition, multiplication, Kronecker sum, tensor product, Hadamard product, and arbitrary real eigenvalue single-matrix functions. We then discuss the composed matrix functions in terms of the estimation of scalar quantities such as inner products, traces, determinants, and Schatten p-norms with bounded errors. We thus provide a framework for compiling instructions for linear algebraic computations into gate sequences on actual quantum computers. The framework for calculating the matrix functions is more efficient than the best classical counterpart for a set of matrices.