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Solution to Problem17


*****

Problem17:

Find an upper triangular matrix A, such that,

> ;

                                 3   [1    30]
                                A  = [       ]
                                     [0    -8]


SOLUTION:


Just multiply a general 2 by 2 upper triangular matrix to see the pattern.

> A := matrix(2,2,[a,b,0,c]);

                                      [a    b]
                                 A := [      ]
                                      [0    c]
> A2 := evalm(A^2);
                                  [ 2             ]
                                  [a     a b + b c]
                            A2 := [               ]
                                  [          2    ]
                                  [0        c     ]
> A3 := evalm(A^3);
                            [ 3     2                  ]
                            [a     a  b + (a b + b c) c]
                      A3 := [                          ]
                            [                3         ]
                            [0              c          ]

Hence, we need to find a,b and c so that A3 coincides with the matrix A^3 given in the problem. The three equations are,

> Eq1 := a^3 = 1; Eq2 := c^3 = -8; Eq3 := b*(a^2+a*c+c^2) = 30;

                                         3
                                 Eq1 := a  = 1

                                        3
                                Eq2 := c  = -8

                                    2          2
                         Eq3 := b (a  + c a + c ) = 30

and the solution is straight forward,

> a := 1: c := -2: Eq3;

                                   3 b = 30

and b = 10. Let's check,

> matrix(2,2,[1,10,0,-2])^3;

                                  [1    10]3
                                  [       ]
                                  [0    -2]

evaluates to:

> evalm(%);

                                   [1    30]
                                   [       ]
                                   [0    -8]

cool!


Link to the commands in this file
Carlos Rodriguez <carlos@math.albany.edu>
Last modified: Wed Feb 9 17:29:14 EST 2000