Suppose a $2 \times 2$ matrix $\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]$ representing a linear transformation maps three points $O(0,0), A(1,0), B(0,1)$ on the coordinate plane to $O(0,0), A'(3, \sqrt{3}), B'(-\sqrt{3}, 3)$ respectively, and maps a point $C(x, y)$ at distance 1 from the origin to point $C'(x', y')$. Select the correct options.
(1) The determinant $\left|\begin{array}{ll} a & b \\ c & d \end{array}\right| = 6$
(2) $\overline{OC'} = 2\sqrt{3}$
(3) The angle between $\overrightarrow{OC}$ and $\overrightarrow{OC'}$ is $60^\circ$
(4) It is possible that $y = y'$
(5) If $x < y$ then $x' < y'$