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I think it depends on the nature of the matrices; it doesn’t make sense to use if your A matrix is just some general dense matrix for example.

Why doesn't it make sense to use in that case?

The dimensions of (A-UBU’) are the same as the dimensions of A. After you apply Woodbury, you have

inv(A) - (other stuff)

So, it only makes sense if A is much easier to solve than (A-UBU’).