You can check that that is indeed a solution by differentiating, now suppose we have a solution we will prove ithas tobe of that form (uniqueness)

Remember that the product rule also holds for Matrixes, we have:

Hence:

and commute, for every matrix .

Thus:

So it is a constant matrix ( because of the 0 derivative) :

Let we get:

And the uniqueness is proven.

For the second part of the question, let

The solutions is of the form:

Let's see the characteristic polynomial:

!!!