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]
|
|
SOLUTION:
| 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. |