TITLE: how to use matrix to prove this identity?
QUESTION [6 upvotes]: if $a_{n},b_{n}$ such $a_{0}=b_{0}=1$ $$\begin{cases}a_{n}=5a_{n-1}+7b_{n-1}\\
b_{n}=7a_{n-1}+10b_{n-1},\forall n=1,2,3,\cdots
\end{cases}$$
show that
$$a_{m+n}+b_{m+n}=a_{m}a_{n}+b_{m}b_{n}$$
It's an interesting identity, and I've proved it with mathematical induction, but I feel like it's more obvious with a matrix, but I don't, so can someone please prove it with a matrix? Thank you
$$\begin{bmatrix}
a_{n}\\
b_{n}\end{bmatrix}=\begin{bmatrix}
5&7\\
7&10\end{bmatrix}\cdot \begin{bmatrix}
a_{n-1}\\
b_{n-1}\end{bmatrix}$$

REPLY [9 votes]: $$
\begin{aligned}
\begin{Bmatrix}
a_{0}&
b_{0}
\end{Bmatrix}\times
\begin{Bmatrix}
a_{m+n}\\
b_{m+n}
\end{Bmatrix}
&=
\begin{Bmatrix}
a_{0}&
b_{0}
\end{Bmatrix}\times
\begin{bmatrix}
5&7\\7&10
\end{bmatrix}^{m+n}\times
\begin{Bmatrix}
a_{0}\\
b_{0}
\end{Bmatrix}\\
&=
\begin{Bmatrix}
a_{0}&
b_{0}
\end{Bmatrix}\times
\begin{bmatrix}
5&7\\7&10
\end{bmatrix}^{m}\times
\begin{bmatrix}
5&7\\7&10
\end{bmatrix}^{n}\times
\begin{Bmatrix}
a_{0}\\
b_{0}
\end{Bmatrix}\\
&=
\left(\begin{bmatrix}
5&7\\7&10
\end{bmatrix}^{m}\times
\begin{Bmatrix}
a_{0}\\
b_{0}
\end{Bmatrix}
\right)^{T}\times
\begin{bmatrix}
5&7\\7&10
\end{bmatrix}^{n}\times
\begin{Bmatrix}
a_{0}\\
b_{0}
\end{Bmatrix}\\
&=\begin{Bmatrix}
a_{m}&b_{m}
\end{Bmatrix}\times
\begin{Bmatrix}
a_{n}\\b_{n}
\end{Bmatrix}
\end{aligned}
$$
This is possible because transposing Your square matrix does not change it.