Question

一个线性代数问题：证明AB-BA不等于E

Answer 1

考虑矩阵的迹.

Tr(AB-BA)=Tr(AB)-Tr(BA)

又因为Tr(AB)=Tr(BA)(因为Tr(AB)=∑aijbji,

Tr(BA)=∑bijaji,所以,Tr(AB)=Tr(BA)),所以

Tr(AB-BA)=0

然而

Tr(E)≠0

所以AB-BA≠E.