ECEF to ECI conversion How to perform the conversion of (r,v) vector from ECEF to ECI (J2000) coordinate frame?

What I tried

I multiplied the matrix ([cosA -sinA 0] [sinA cosA 0] [0 0 1]) by the state vector, where A is Greenwitch mean sidereal time.

This equation shows that the Z axis doesn't change. However, I have the ephemeris of a real satellite in both coordinate frames, and there Z axis differs! 
Where is the issue?
A reference frame that has its Z axis fixed with respect to the Earth's rotation axis is not inertial. The Earth's rotation axis undergoes precession and nutation, and on a shorter term, undergoes polar motion as well.