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Solution to Problem9


*****

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 ]

Link to the commands in this file
Carlos Rodriguez <carlos@math.albany.edu>
Last modified: Tue Feb 8 21:39:06 EST 2000