General Formula: Matrix Inversion Lemma

Let , , and be non-singular square matrices; then

General Formula: Matrix Inversion in Block form

Let a matrix be partitioned into a block form:

where the matrix and matrix are invertible. Then we have

It can be proved that the above two matrix expressions for are equivalent.

Special Case 1

Let a matrix be partitioned into a block form:

Then the inverse of is

where .

Special Case 2

Suppose that we have a given matrix equation

 (1)

where and are invertible matrices and all matrices are of compatible dimensions in the above equation. Please find the matrices and in terms of the given matrices , , , and .