Proving matrix exponential

Question

Can anyone tell me how the following is derived?

where $A$ is a matrix.

Answer 1

Write

$\dot x = Ax \tag 1$

as

$\dot x - Ax = 0; \tag 2$

multiply through by $e^{-At}$:

$e^{-At} \dot x - e^{-At}Ax = 0; \tag 3$

note that

$e^{-At}A = Ae^{-At}, \tag{3.5}$

and re-arrange (3):

$e^{-At} \dot x - Ae^{-At}x = 0; \tag 4$

observe that the Leibniz rule for differentiating products yields

$\dfrac{d}{dt} (e^{-At} x) = -Ae^{-At}x + e^{-At} \dot x = e^{-At} \dot x - Ae^{-At}x = 0; \tag 5$

thus

$e^{-At} x = c, \; \text{a constant vector}, \tag 6$

whence

$x(t) = e^{At}c; \tag 7$

taking $t = 0$ yields

$x(0) = e^{A \cdot 0}c = e^0c = Ic = c; \tag 8$

thus (7) becomes

$x(t) = e^{At}x(0). \tag 9$