# linear algebra – Implementing solution of a system of recurrence relations

Let us consider a system of recurrence relations such as

$$a_{n-1} = ( lambda_1 + n lambda_2 ) a_n + lambda_3 b_n$$

$$b_{n-1} = ( lambda_4 + n lambda_2 ) b_n + lambda_5 a_n$$

subject to the initial conditions say, $$a_0 = k_1$$ and $$b_0 = k_2$$. So, what I have done so far is that I reexpressed the system as

$$v_n = A^T v_{n-1}$$

or equivalently,

$$v_n = (A^T)^n v_0$$,

where $$v_n = (a_n, b_n)^T$$, and $$A$$ is the coefficient matrix including $$n$$.

How do I implement such a system in Mathematica to get the sequences $$a_n$$ and $$b_n$$?

Also, is there a function, in Mathematica, similar to RSolve for solving such systems?