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Solution to Problem8


*****

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.


Link to the commands in this file
Carlos Rodriguez <carlos@math.albany.edu>
Last modified: Tue Feb 8 21:26:17 EST 2000