A New Generalization of the Liouville-Jacobi Identity

A generalized Liouville-Jacobi Identity is proved for the determinant $\det X(t)$ of a solution $X(t)$ to the linear nonhomogeneous first-order matrix differential equation with left- and right-coefficient matrices $\frac{\mathrm{d}}{\mathrm{d}t}X(t)+A(t)X(t)+X(t)B(t)=F(t),X(t_0)=X_0\mathrm{.}$