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

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.

Let a
matrix
be partitioned into a block form:

Then the inverse of is

where .

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 .