Calculate matrix to power 101

Question

Being given that

A = $\bigl( \begin{smallmatrix} 1 \\ 1 \\ -1 \end{smallmatrix} \bigr)$ and B = $\bigl( \begin{smallmatrix} 1 &2 &-1 \end{smallmatrix} \bigr)$,

Calculate C = $A \cdot B$ and $C^{101}$.

Answer 1

$$A.B=\begin{pmatrix}1 &2&-1\\1&2&-1&\\-1&-2&1\end{pmatrix}$$

$$\begin{pmatrix}x&2x&-x\\x&2x&-x\\-x&-2x&x\end{pmatrix}\begin{pmatrix}x&2x&-x\\x&2x&-x\\-x&-2x&x\end{pmatrix}=4\begin{pmatrix}x^2&2x^2&-x^2\\x^2&2x^2&-x^2\\-x^2&-2x^2&x^2\end{pmatrix}$$

now, $$C^2=4\begin{pmatrix}1&2&-1\\1&2&-1\\-1&-2&1\end{pmatrix}$$$$C^3=16\begin{pmatrix}1&2&-1\\1&2&-1\\-1&-2&1\end{pmatrix}$$$$\vdots$$$$C^{n+1}=4^n\begin{pmatrix}1&2 &-1\\1&2&-1\\-1&-2&1\end{pmatrix}$$

Hence, $$C^{101}=4^{100}\begin{pmatrix}1 &2&-1\\1&2&-1&\\-1&-2&1\end{pmatrix}=4^{100}C$$

Answer 2

Hint : $C^{101}=A \cdot (B \cdot A)^{100} \cdot B$