How to solve dp with fast matrix exponentiation when matrix is not fixed?

I am trying to solve www.codechef.com/ACMKAN14/problems/ACM14KN2. It requires dp with fast matrix exponentiation but the matrix doesn’t seem to be fixed. When k=0 the recurrence is F[n] = F[n-1] + F[n-2] but I can’t figure out the complete solution. How can the matrix be generalized?

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