Does $A$ commute with $e^{\int A \: dt}$

Question

I have been studying the linear system of the form:

$$D_tX = AX + \textbf{b}$$

Where $A$ is not necessarily constant

Suppose we aim to find an integrating factor $M$ such that:

$$M[D_tX - AX] = D_t(MX)$$

This gives:

$$MD_tX - MAX = (D_tM)X + M(D_tX)$$

By equating coefficients we get:

$$D_tM = -MA$$

Solving this gives:

$$M = e^{-\int A \: dt}$$

But

$$D_t(e^{-\int{A} \: dt}) = -Ae^{-\int{A} \: dt} = -AM$$

So can we conclude that these two matrices commute?

edit

I have proven that

$$AM = MA$$

if and only if

$$A\left(\int{A} \: dt \right ) = \left (\int{A} \: dt \right ) A$$

edit 2

After looking further into the question, it appears that for non-constant matrices

$$D_te^{A(t)} \neq \left ( D_tA(t) \right ) e^{A(t)}$$

more can be found here

Answer 1

Let $$M = \left(e^{-\int A^T\, dt}\right)^T$$

Then $$D_t M = \left(D_t e^{-\int A^T\, dt}\right)^T = \left(-A^Te^{-\int A^T\, dt}\right)^T = \left(-A^TM^T\right)^T = -MA$$