Problem9:Use the result of the previous problem to find the inverse of the matrix, |
> ;
[ 1 1 0 0]
[ ]
[-1 1 0 0]
[ ]
[ 1 1 1 1]
[ ]
[ 1 1 -1 1]
|
|
SOLUTION:
| We partition the matrix into, |
> ;
[A 0]
[ ]
[B A]
| where, |
> A := matrix(2,2,[1,1,-1,1]); B := matrix(2,2,[1,1,1,1]);
[ 1 1]
A := [ ]
[-1 1]
[1 1]
B := [ ]
[1 1]
| as it was seen in Problem8 the inverse is also partitioned into 4 blocks with the inverse of A in the main diagonal, 0 on the upper right hand side and C in the lower left hand side. The matrix C is given by, |
> C := - inverse(A) &* B &* inverse(A);
//[1/2 -1/2] \ [1/2 -1/2]\
C := -||[ ] &* B| &* [ ]|
\\[1/2 1/2 ] / [1/2 1/2 ]/
> evalm(C);
[ 0 0]
[ ]
[-1 0]
| Thus, the inverse of the original 4x4 matrix is, |
> ;
[1/2 -1/2 0 0 ]
[ ]
[1/2 1/2 0 0 ]
[ ]
[ 0 0 1/2 -1/2]
[ ]
[-1 0 1/2 1/2 ]