# Linear Algebra – Blockwise inversion of the matrix with rectangular blocks

Given a matrix in block form:

$$W = left ( begin {array} {cc} A & B \ C & D end {array} right)$$

from where $$A, D$$ Are square matrices, we can write the inverse in the same block form:

$$W ^ {- 1} = left ( begin {array} {cc} A ^ {- 1} + A ^ {- 1} B (D – CA ^ {- 1} B) ^ {- 1} CA ^ {- 1} & – A ^ {- 1} B (DC A ^ {- 1} B) ^ {- 1} \ – (D – C A ^ {- 1} B) ^ {- 1} C A ^ {- 1} & (D – C A ^ {- 1} B) ^ {- 1} end {array} right)$$

provided that $$A$$ and $$D-CA ^ {- 1} B$$ are invertible.

Is there a more general version of the block inversion that allows it? $$A$$ or $$D$$ be rectangular?