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Solution to Problem7


*****

Problem7:

Find the inverse of

> ;

                              [cos(t)     sin(t)]
                              [                 ]
                              [-sin(t)    cos(t)]


SOLUTION:


Recall the formula for the inverse of a 2x2 matrix,

> A := matrix(2,2,[a,b,c,d]); A1 := inverse(A);

                                      [a    b]
                                 A := [      ]
                                      [c    d]

                            [     d               b    ]
                            [ ---------     - ---------]
                            [ a d - b c       a d - b c]
                      A1 := [                          ]
                            [      c             a     ]
                            [- ---------     --------- ]
                            [  a d - b c     a d - b c ]

in words: flip the main diagonal, change the signs of the second diagonal and divide through by the determinant (ad-bc). For the matrix given in the problem the determinant is,

> ;

                                   2         2
                             cos(t)  + sin(t)  = 1

so the inverse is given by:

> ;

                              [cos(t)    -sin(t)]
                              [                 ]
                              [sin(t)    cos(t) ]

NOTE: pre multiplication by the matrix given in the problem rotates the coordinates of a vector in an angle of t radians counter clockwise. Hence, the inverse of this transformation is just rotation in -t (i.e. same angle but now clockwise). Replacing t by -t in the original matrix gives the inverse as it should.


Link to the commands in this file
Carlos Rodriguez <carlos@math.albany.edu>
Last modified: Tue Feb 8 20:53:49 EST 2000