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