System of linear differential equations

Given System of linear differential equations with constant coefficient:

The matrix coefficient :

And also given that the vector :

Is a solution to the system.
1.Show that x1(t),x2(t),...,xn(t) are differetiable from every order in interval I and also that:
$\frac{d^k \mathbf{x}(t)}{dt^k} = \mathbf{A}^k \mathbf{x}(t),$
2.Show that there Is System of linear differential equations with constant coefficient from order n such that
x1(t),..,xn(t) are a solution. Hint: Kylie Hamilton theorem

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• I understand why we can differentiate one time, but what is the justification for second, third, and go on?

• It's by mathematical induction. The step from n-1 to n allows to go from 2 to 3 to 4 to ... To be rigorous you only need the step from 1 to 2 (the base case) and from n-1 to n (the induction step).

The answer is accepted.
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