What ? You couldn’t understand my clear and concise english?
When you Add an Object the position in 3D-space it is added is stored in the OB:Data as the ‘Origin’. (for this example say we parent a cube[child] to a plane [parent[) If you use Ctrl-P then the origin of the Child will be retained. So when you terminate the parented relationship the Child can return to the original position of its Origin.
If you use Shft-Ctrl-P the Origin is reset to the Origin of the Parent, the Plane.
Now, when you “Clear Parent Inverse” you would expect the same thing to happen as happens with Shft-Ctrl-P, but it doesn’t; the child does move, but not to Object Center of the Parent as you would expect. The reason for this is that the child maintains it’s spatial arrangement with the parent because that is usually desireable (think of Duplis) but gives up it’s link to its own spacial data.
If you unparent everything, Ctrl-A on both objects then rotate the plane (will be parent again) 90.deg and then parent them again, you’ll see that the child maintains it’s spatial integrity. Alt-P and “Clear Parent Inverse” and the Child is now subject to the spatial data of the Parent.
I have no idea what Inverse Matrix means.