$n$ 次正方行列 $A, B$ に対して、$[A, B] = AB – BA$ とおき、これを $A$ と $B$ の交換子積という。交換子積に関する次の性質を示せ。
(1) $[A, B] = – [B, A]$
(2) $[A, A] = 0$
(3) $[[A, B], C] + [[B, C], A] + [[C, A], B] = O$
(1)
\begin{align}
[A, B] &= AB – BA \\
&= -(BA – AB) \\
&= – [B, A]
\end{align}
(2)
\begin{align}
[A, A] &= A A – A A \\
&= O
\end{align}
(3)
\begin{align}
[[A, B], C] + [[B, C], A] + [[C, A], B] &=[AB – BA, C] + [BC – CB, A] + [CA – AC, B] \\
&= ABC – BAC – CAB + CBA + BCA – CBA – ABC + ACB + CAB – ACB – BCA + BAC \\
&= O
\end{align}
