Conder the matrix

A=[cos B -sin Q

sin B cos B ]

Multiplying a vector x in R^2 by A has the effect of rotationg x by angle B counter-clockwise about the origin.

1. Reasoning geometrically, give two values of B in [ 0,2pi) for which A has real eigenvalues. For each such B, state the eigenvalues and corresponding eigenvalues associated with them.

2.There is a value of B in [0,pi] for which A has i as an eigenvector. Find this value of B and find an eigenvector corresponding to the eigenvalue i in this case.

3. The product of the eigenvalues of a matrix are always equal to the determinant of the matrix. Use this fact to determine the other eigenvalue of A for the value of B u find in part 2. Find a corresponding eigenvector.