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Solution to Problem11


*****

Problem11:

Consider the matrices,

> ;

                                  [3     4     1]
                                  [             ]
                             A := [2    -7    -1]
                                  [             ]
                                  [8     1     5]
> ;
                                  [3     4     1]
                                  [             ]
                             B := [2    -7    -1]
                                  [             ]
                                  [2    -7     3]

Find elementary matrices E and F such that,

> ;

                                    E A = B

                                    F B = A


SOLUTION:


We only need to identify the elementary row operation that takes A into B. By inspection we see that B is obtained from A by adding (-2) times the first row to the third row. The matrix representation E of an elementary row operation is obtained by applying the corresponding transformation to the identity matrix since, EI=E. Hence,

> E := addrow( diag(1,1,1), 1,3,-2);

                                   [ 1    0    0]
                                   [            ]
                              E := [ 0    1    0]
                                   [            ]
                                   [-2    0    1]

let's check,

> A := matrix(3,3,[3,4,1, 2,-7,-1, 8,1,5]):
> evalm( E &* A );

                                [3     4     1]
                                [             ]
                                [2    -7    -1]
                                [             ]
                                [2    -7     3]

so yep! we got B. It works. Now let's do the trick in reverse, i.e. go from B to A with an elementary row operation. Clearly we get A when we add to the third row of B, (2)times its first row. So we have,

> F := addrow( diag(1,1,1), 1,3, 2);

                                   [1    0    0]
                                   [           ]
                              F := [0    1    0]
                                   [           ]
                                   [2    0    1]

Now notice that F is the inverse of E as it should, right?

> evalm( E &* F );

                                 [1    0    0]
                                 [           ]
                                 [0    1    0]
                                 [           ]
                                 [0    0    1]

Did you make sure you understand the maple commands in this problem? Check the maple help on: addrow, diag, &* and evalm....


Link to the commands in this file
Carlos Rodriguez <carlos@math.albany.edu>
Last modified: Wed Feb 9 10:54:39 EST 2000