# how to use matrix to prove this identity?

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 $$\begin{bmatrix} a_{n}\\ b_{n}\end{bmatrix}=\begin{bmatrix} 5&7\\ 7&10\end{bmatrix}\cdot \begin{bmatrix} a_{n-1}\\ b_{n-1}\end{bmatrix}$$

• I am curious about Your induction proof too, could You put it here too? Commented Mar 24, 2020 at 6:17

\begin{aligned} \begin{Bmatrix} a_{0} \\ b_{0} \end{Bmatrix}^{\top} \begin{Bmatrix} a_{m+n}\\ b_{m+n} \end{Bmatrix} &= \begin{Bmatrix} a_{0} \\ b_{0} \end{Bmatrix}^{\top} \begin{bmatrix} 5&7\\7&10 \end{bmatrix}^{m+n} \begin{Bmatrix} a_{0}\\ b_{0} \end{Bmatrix}\\ &= \begin{Bmatrix} a_{0} \\ b_{0} \end{Bmatrix}^{\top} \begin{bmatrix} 5&7\\7&10 \end{bmatrix}^{m} \begin{bmatrix} 5&7\\7&10 \end{bmatrix}^{n} \begin{Bmatrix} a_{0}\\ b_{0} \end{Bmatrix}\\ &= \left(\begin{bmatrix} 5&7\\7&10 \end{bmatrix}^{m} \begin{Bmatrix} a_{0}\\ b_{0} \end{Bmatrix} \right)^{T} \begin{bmatrix} 5&7\\7&10 \end{bmatrix}^{n} \begin{Bmatrix} a_{0}\\ b_{0} \end{Bmatrix}\\ &=\begin{Bmatrix} a_{m} \\ b_{m} \end{Bmatrix}^{\top} \begin{Bmatrix} a_{n}\\b_{n} \end{Bmatrix} \end{aligned}
• Symmetry in the matrix and starting at $(1,1)$ help Commented Mar 24, 2020 at 13:51