<s>[INST] Finding n-th term of a matrix I have to find the n-th power of the following matrix $$A=\begin{pmatrix} 2 & 0&1\\ 0 & -3& 0 \\ 1 & 0&2 \end{pmatrix}
$$ I computed a few powers $$A^2=\begin{pmatrix} 4+1& 0&4\\ 0 & 9& 0 \\ 4 & 0&1+4\end{pmatrix}$$
$$A^3 =\begin{pmatrix} 14 & 0&13\\ 0 & -27& 0 \\ 13& 0&14\end{pmatrix}$$
$$A^n=\begin{pmatrix} \frac12(3^n+1) & 0&\frac12(3^n-1)\\ 0 &( -3)^n& 0 \\ \frac12(3^n-1) & 0&\frac12(3^n+1)\end{pmatrix}$$  And is just induction left which is easy, but I found by check and try the n-th term, is it possible to find it in a proper way? 
 [/INST] No, that is not true, we can also use other way : 
$A=\begin{pmatrix} 2 & 0&1\\ 0 & -3& 0 \\ 1 & 0&2 \end{pmatrix}$
$A=B+C$
$C=\begin{pmatrix}  0& 0&1\\ 0 & 1& 0 \\ 1 & 0&0 \end{pmatrix}$
$B=\begin{pmatrix} 2 & 0&0\\ 0 & -4& 0 \\ 0 & 0&2 \end{pmatrix}$
$A^n = (B+C)^n = \sum_{k=0}^{n} C_n^k B^k C^{n-k} $
$= \frac{1}{2} \sum_{k=0}^{n} C_n^k \begin{pmatrix} 2^{n-k} & 0&0\\ 0 & (-4)^{n-k} & 0 \\ 0 & 0&2^{n-k} \end{pmatrix} \begin{pmatrix} (-1)^k+1 & 0&(-1)^{k+1}+1\\ 0 & 2& 0 \\ (-1)^{k+1}+1 & 0&(-1)^k+1 \end{pmatrix} $
$=\begin{pmatrix} \frac12(3^n+1) & 0&\frac12(3^n-1)\\ 0 &( -3)^n& 0 \\ \frac12(3^n-1) & 0&\frac12(3^n+1)\end{pmatrix}$
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