TITLE: show that $\det{(A^2+B^2+C^2-BA-CB-AC)}=0$
QUESTION [5 upvotes]: Let $A,B,C\in M_{2}(C)$ such that
$$A^2+B^2+C^2=AB+BC+CA$$
show that
$$\det{(A^2+B^2+C^2-BA-CB-AC)}=0$$
from:matrix indentity

REPLY [3 votes]: Hint :
if $A\in M_{2\times 2}(\mathbb{R})$ then :
$$\det(A)=(tr(A)^2-tr(A^2))/2,$$
and we know that $$tr(A^2+B^2+C^2-BA-CB-AC)=tr(AB+BC+CA-BA-CB-AC)=0$$