To get i as an eigenvalue (NOT eigenvector) we must be thinking of (x,y) as representing the complex number x+iy. Av= iv becomes A(x+iy)= i(x+iy). What is i(x+iy)? What point does that correspond to? What angle is (x,y) rotated through?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.
All rotations have the same determinant. What is it? (Hint: the identity transformation:x-> x is a rotation through 0 degrees.)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.