Hi,

in the attached file you can find a hint how to calculate the inverse of

your matrix (up to a scalar coefficient). In general, it is a symmetric tridiagonal.

The proof (not using Wolfram) is based on direct calculations.

Thank you Yuri S Semenov sir for hint. can you please explain on how the matrix becomes banded i.e. how some elements become zero.

Can you please suggest me any reference books where I can get a more insight on this problem.

Thanks once again sir

I've noticed (using Wolfram for small dimensions t=4 and 6) that the inverse matrix is of special kind (tridiagonal). It remained only to prove that in general it is also of this kind. The direct calculations confirmed this.

In my algebraic experience I had to invert some parametrically given matrices. Sometimes I guessed the answer and then proved that the result was correct.

I would suggest you to invert the found tridiagonal inverse matrix in order to obtain the original matrix. I'm short of references (on the inversion of tridiagonal matrices) but maybe this one will help https://arxiv.org/pdf/1803.04516v2.pdf

Thank You very much Yuri S Semenov sir for the help.