transferring coordinate systems

Hello,

So there is a 'world coordinate system' in normal 3D space 'xyz'.
If I apply a rigid-body transformation to this coordinate system I get a new coordinate system, (A).
By 'rigid-body transformation' I mean first applying a 3X3 matrix R_1 on any point in world coordinate system and then translating it by 3-vector T_1.
Now imagine I apply a different 'rigid-body transformation' on the same global coordinate system , denoted by rotation matrix R_2 and translation vector T_2, to get a new coordinate system (B).

Now my question is what Rotation/Translation will transform coordinate system (A) to (B)?
will it be the following?
Rotation = R_2 X (inverse of R_1)
Translation = T_2 - R_2 X (inverse of R_1) X T_1

There is this software I am using and it needs these rotation and translation matrix/vector. But above values don't work!

Thanks for your help.

Assuming that R_2, and R_1 are linearly indepdent. If for any . Since R_1 is lin independent, we are assured some x(unique) exists in our global coordinate system. So , again an inverse also exists for , call it . So . So, x is in our global coordinate sys and we know how to get from our global coordinates to B, so , so to get from A to B, we have will do the transformation