In this paper, we construct and prove the existence of theoretical solutions to non-isentropic Euler equations with a time-dependent linear damping and Coriolis force in Cartesian form. New exact solutions can be acquired based on this form with examples presented in this paper. By constructing appropriate matrices $A(t)$, and vectors $\mathbf{b}(t)$, special cases of exact solutions, where entropy $s=\ln \rho$, are obtained. This is the first matrix form solution of non-isentropic Euler equations to the best of the authors' knowledge.