행렬 거듭제곱의 합

행렬 $\displaystyle A = {\begin{pmatrix} 0 & & 1 \\ 1 & & 0 \end{pmatrix}}$에 대하여, $\displaystyle A ^{2} = {\begin{pmatrix} 0 & & 1 \\ 1 & & 0 \end{pmatrix}} {\begin{pmatrix} 0 & & 1 \\ 1 & & 0 \end{pmatrix}} = {\begin{pmatrix} 1 & & 0 \\ 0 & & 1 \end{pmatrix}}$ $\displaystyle A ^{3} = A ^{2} A = {\begin{pmatrix} 1 & & 0 \\ 0 & & 1 \end{pmatrix}} {\begin{pmatrix} 0 & & 1 \\ 1 & & 0 \end{pmatrix}} = {\begin{pmatrix} 0 & & 1 \\ 1 & & 0 \end{pmatrix}} = A$ 즉, $A ^{2 n} = E , A ^{2 n - 1} = A$이므로, $\displaystyle \sum\limits _{n=1} ^{2002} A ^{n} = A + A ^{2} + A ^{3} + \cdots + A ^{2002}$ $= A + E + A + \cdots + E$ $= 1001 A + 1001 E$ $\displaystyle = 1001 {\begin{pmatrix} 0 & & 1 \\ 1 & & 0 \end{pmatrix}} + 1001 {\begin{pmatrix} 1 & & 0 \\ 0 & & 1 \end{pmatrix}} = 1001 {\begin{pmatrix} 1 & & 1 \\ 1 & & 1 \end{pmatrix}}$ 따라서, $\displaystyle \frac{1}{2002} \sum\limits _{n = 1} ^{2002} A ^{n} = \frac{1}{2002} \cdot 1001 {\begin{pmatrix} 1 & & 1 \\ 1 & & 1 \end{pmatrix}} = \frac{1}{2} {\begin{pmatrix} 1 & & 1 \\ 1 & & 1 \end{pmatrix}} = {\begin{pmatrix} a & & b \\ c & & d \end{pmatrix}}$ 따라서, $a + b + c + d = 2$이다.