Variable matrices
If I type
sage: M = matrix(SR, 2, var('a,b,c,d'))
sage: show(~M)it gives me the inverse of the matrix M with entries in terms of a, b, c, d.
Now what I want is to set four variable matrices A, B, C, D, each of size 2x2,
and then I want to create the block matrix
sage: N = block_matrix([[A, B], [C, D]])Finally I want some functions of N in terms of A, B, C, D.
For example, how can I get det(N) in terms of A, B, C, D?
You want det in terms of \det A, ..., \det D? SageMath is unable to give you this to the best of my knowledge. Note that it is not suprising: There is no such expression in a very general case. Only if some matrix is known to be invertible, or some matrices commute, etc. See on Wikipedia.
Now in your case of 2\times 2 blocks, you may of course define 16 variables a_{11}, ..., a_{22}, ..., d_{22} and ask for this 16\times 16 determinant. Then, you can try to reconstruct some expression based on the four 2\times 2 determinants. But you'll need to have some assumptions at some point I guess.