If there is a problem like this:↵
Give $4$ integer $a, b, c, d$, $u_n=au_{n-1}+bv_{n-1}, v_n=cu_{n-1}+dv_{n-1}$, we knew $u_0, v_0$. We need to calculate $u_n$ with $n\le 10^9$.↵
This problem can be solved by using matrix exponentiation, we have $u_n=(a+d)u_n-(ad-bc)u_{n-1}$ with $n>1$.↵
↵
[My fast proof](https://hackmd.io/VlVjQ2rTTxaVJRcKhb6hvw)
Give $4$ integer $a, b, c, d$, $u_n=au_{n-1}+bv_{n-1}, v_n=cu_{n-1}+dv_{n-1}$, we knew $u_0, v_0$. We need to calculate $u_n$ with $n\le 10^9$.↵
This problem can be solved by using matrix exponentiation, we have $u_n=(a+d)u_n-(ad-bc)u_{n-1}$ with $n>1$.↵
↵
[My fast proof](https://hackmd.io/VlVjQ2rTTxaVJRcKhb6hvw)
