using matrices to propagate a child bone pose back to its parent?

I don’t know if this image helps explain at all what I’m trying to do:

assuming figure 1 is the child parent bone relationship in rest pose, I want to be able to transform the child like figure 2 for example, and then have the parent align itself like in figure 3, so if the child’s transform was set back to rest, the child would visually appear the same as in figure 2.

My thought was to multiply the child bone’s matrix with the inverse of its matrix_local (which I understand to be its rest pose relative to its parent) to work backwards to get the parent’s matrix? Am I way off base? Is there some easier way to do this?