Let A, B, and 0 be 2x2 matrices. Assuming that A is invertible, find a matrix C such that
[A^-1|0 ]
[C |A^-1]
is the inverse of the partitioned matrix
[A|0]
[B|A]
To find the matrix C such that
```
[A^-1|0]
[C |A^-1]
```
is the inverse of the partitioned matrix
```
[A|0]
[B|A]
```
we need to use the concept of block matrix inversion.
First, we'll write the given partitioned matrix as a block matrix:
```
M = [A | 0]
[B | A]
```
The inverse of M is given by:
```
M^(-1) = [X | Y]
[Z | W]
```
To find the matrix X, Y, Z, and W, we can use the formulas for block matrix inversion.
The formula for X is:
```
X = (A - B(A^-1)0)^(-1)
```
Since 0 is a zero matrix, the term B(A^-1)0 becomes zero, and X simplifies to A^(-1).
```
X = A^(-1)
```
The formula for Z is:
```
Z = -X(BA^-1)^(-1)
```
Let's calculate BA^-1:
```
BA^(-1) = [B | A] * [A^-1 | 0]
= [BA^-1 | A]
```
Therefore, Z simplifies to:
```
Z = -X(BA^-1) = -A^(-1)(BA^-1) = -(A^(-1)(BA^-1))
```
Similarly, Y and W can be calculated using the following formulas:
```
Y = X(BA^-1)^(-1)0
W = -(A^-1)(BA^-1)A
```
Substituting the values we calculated earlier:
```
Y = A^(-1)(BA^(-1))0 = 0
W = -(A^-1)(BA^-1)A
```
Putting everything together:
```
M^(-1) = [X | Y] = [A^-1 | 0]
[Z | W] [-(A^-1)(BA^-1) | 0]
```
Therefore, the matrix C can be represented as:
```
C = -(A^-1)(BA^-1)
```
This is the matrix C such that [A^-1|0 ] [C |A^-1] is the inverse of the given partitioned matrix [A|0] [B|A].