# Question #0ab0c

Aug 5, 2017

See below.

#### Explanation:

Taking

${C}_{1} = \left(\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right)$
${C}_{2} = \left(\begin{matrix}0 & 0 & 0 \\ 4 & 0 & 0 \\ 8 & 4 & 0\end{matrix}\right)$
${C}_{3} = \left(\begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ 8 & 0 & 0\end{matrix}\right)$

we have

${A}^{n} = {C}_{1} + n {C}_{2} + {n}^{2} {C}_{3}$

NOTE:

The $A$ characteristic polynomial is

${s}^{3} - 3 {s}^{2} + 3 s - 1 = {\left(s - 1\right)}^{3} = 0$

and this polynomial is such that

${A}^{3} - 3 {A}^{2} + 3 A - {I}_{3} = {0}_{3}$

so the matrix obeys the recurrence equation

${A}^{n} - 3 {A}^{n - 1} + 3 {A}^{n - 2} - {A}^{n - 3} = 0$

which has the solution

${A}^{n} = {C}_{1} + n {C}_{2} + {n}^{2} {C}_{3}$