Problem7:Find the inverse of 
> ;
[cos(t) sin(t)] [ ] [sin(t) cos(t)]

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 (adbc). 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. 