Analytical formula of the elements of the inverse of a matrix plus another matrix.

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I have 2 Fisher information matrices A and B. I have analytical formula of elements of matrix B (let's say B_ij).

Now, I would like to find an expression of elements (k,l) of the inverse matrix (A+B)^-1, i.e the elements ((A+B)^-1)_kl. The dual space corresponds to the covariance matrix.

The fisher information matrix A is known numerically.

Do you think it is possible ?

ps : the elements that could interest me the most should be the (k,l)=(i,j) elemements where (i,j) are the analytical terms that I know on matrix B.