AB=BA 충분조건 보기

ㄱ. $A + B = 2 E$ 이면 $B = 2 E - A$이므로 $AB = A ( 2 E - A ) = 2 A - A ^{2}$$= ( 2 E - A ) A = BA$ (참) ㄴ. [반례] $\displaystyle A = \left( \begin{array}{ll} \begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix} \end{array} \right) , B = \left( \begin{array}{ll} \begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix} \end{array} \right)$이면 $A ^{2} = E$ 이므로 $A ^{2} B = BA ^{2} = B$이지만 $\displaystyle AB = \left( \begin{array}{ll} \begin{matrix} 1 & 1 \\ 1 & 0 \end{matrix} \end{array} \right) \ne \left( \begin{array}{ll} \begin{matrix} 0 & 1 \\ 1 & 1 \end{matrix} \end{array} \right) = BA$ (거짓) ㄷ. $A ^{2} B = A + E$에서 $A ^{2} B - A =$$A ( AB - E ) = E$ … ㉠ $A ^{- 1} = AB - E$에서 $( AB - E ) A = E$, $ABA - A = E$ … ㉡ ㉠, ㉡에서 $A ^{2} B = ABA$이고 $A ^{- 1}$을 곱하면 $AB = BA$ (참)