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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#