All computations take place over .
Let , , be differentiable functions of the real variable which are related as follows: . Letting and letting , we can write this system as . Any matrix whose columns are linearly independent solutions of this equation is called a fundamental matrix of the equation. (From the theory of differential equations, we know that the solutions of this equation are an -dimensional -vector space, of which the columns of a fundamental matrix form a basis.)
Prove that is a fundamental matrix of . Show also that if is an constant matrix (i.e., entries not dependent on ) then is the particular solution of this equation satisfying .
Recall from this previous exercise that is nonsingular, and so its columns are linearly independent. Now write as a column matrix. From this previous exercise, we have , and so . In particular, the columns of are solutions of . So by definition is a fundamental matrix of .
Now let . Certainly , and using part 2 of this previous exercise, we have .