In the following B = (1, 2, 0, 1) , B-1 = (1, -2, 0, 1) and C = (1, 0, 0, -1)in the following order for all three matrices (top left, top right, bottom left, bottom right).

My question is:

If P' is the image of P under a matrix D = BCB-1. If P is not on the x-axis, why is PP' bisected by the x-axis and is at a constant angle to the x-axis, for any choice of P?

I can visually see what's happening but how can I show it? Please help