# 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>