행렬 거듭제곱의 합

$a _{11} = 0 , a _{12} = - 1 , a _{21} = 1 , a _{22} = 0$이므로 $\displaystyle A = \begin{pmatrix} 0 & - 1 \\ 1 & 0 \end{pmatrix}$ $\displaystyle A ^{2} = \begin{pmatrix} 0 & - 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 0 & - 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} - 1 & 0 \\ 0 & - 1 \end{pmatrix} = - E$ $A ^{3} = - A , A ^{4} = E , A ^{5} = A , \cdots$ $\therefore A + A ^{2} + A ^{3} + \cdots + A ^{2010} = ( A + A ^{2} + A ^{3} + A ^{4} ) + \cdots$ $+ ( A ^{2005} + \cdots + A ^{2008} ) + A ^{2009} + A ^{2010}$ $= O + \cdots + O + A + A ^{2}$ $\displaystyle = A - E = \begin{pmatrix} - 1 & - 1 \\ 1 & - 1 \end{pmatrix}$ 따라서 $( 2 , 1 )$의 성분은 1이다.