matrix問題

Let A=[a b,c d] B=[e f, g h]
f(A+B)=a+d+e+h
f(A)+f(B)=a+d+e+h
So f(A+B)=f(A)+f(B)

f(AB)=f(ae+bg af+bh,ce+dg cg+dh)=ae+bg+cf+dh
f(BA)=f(ea+fc eb+fd, ga+hc gb+hd)=ea+fc+gb+hd
So f(AB)=f(BA)

PQ-QP=[1 2, 3 4]
Since f(1 2,3 4)=5 and f(PQ-QP)=f(PQ)-f(QP)=0 not equal to 5
So there are no two 2x2 matrices P and Q such that PQ-QR =[1 2, 3 4]