Problem8:Let A, B, and 0 be 2x2 matrices. Assuming that A is invertible, find a matrix C so that, 
> ;
[ 1 ] [A 0 ] [ ] [ 1] [ C A ]
is the inverse of the partitioned matrix 
> ;
[A 0] [ ] [B A]

We must have, 
> ;
[ 1 ] [A 0] [A 0 ] [I 0] [ ] [ ] = [ ] [B A] [ 1] [0 I] [ C A ]
so the second row times first column gives the equation for C, 
> ;
1 B A + A C = 0 1 A C =  B A 1 1 C =  A B A
Remember this, you will need it for the next problem. Also notice that there is nothing special about the matrices being 2x2. The same is true for nxn matrices. 