That is partially correct. To have that form, the matrix must be in thespecialunitary group SU(2). This means that in addition to being unitary, the matrix has determinant 1.

If the matrix is unitary then . Compare matrix elements on both sides of that equation, together with the equation ad-bc=1, and you will find that and . (I'm using bars rather than stars to denote complex conjugates.)

The general unitary matrix could have determinant –1, in which case it has the form .