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. |