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Answer 1 · 2017-08-05T13:19:21

See below.

Taking

$C_1=((1,0,0),(0,1,0),(0,0,1))$
$C_2=((0,0,0),(4,0,0),(8,4,0))$
$C_3=((0,0,0),(0,0,0),(8,0,0))$

we have

$A^n = C_1+n C_2+n^2C_3$

NOTE:

The $A$ characteristic polynomial is

$s^3-3s^2+3s-1=(s-1)^3 = 0$

and this polynomial is such that

$A^3-3A^2+3A-I_3=0_3$

so the matrix obeys the recurrence equation

$A^n-3A^(n-1)+3A^(n-2)-A^(n-3)=0$

which has the solution

$A^n = C_1+n C_2+n^2C_3$