Let be a linear system of first order differential equations over as in this previous exercise. Suppose is a fundamental matrix of this system (i.e. a matrix whose columns are linearly independent solutions) and let be a nonsingular matrix over . Prove that is also a fundamental matrix for this differential equation.
Write as a column matrix. Now by the definition of our matrix derivative and part 2 of this previous exercise. This is then equal to . In particular, the columns of are solutions of . Moreover, since is nonsingular, is nonsingular. So is also a fundamental matrix.