행렬의 거듭제곱 A^2013

$\displaystyle A = \begin{pmatrix} {0} & {1} \\ {- 1} & {1} \end{pmatrix}$ $A ^{2}$$\displaystyle = \begin{pmatrix} {0} & {1} \\ {- 1} & {1} \end{pmatrix} \begin{pmatrix} {0} & {1} \\ {- 1} & {1} \end{pmatrix}$$\displaystyle = \begin{pmatrix} {- 1} & {1} \\ {- 1} & {0} \end{pmatrix}$ $A ^{3}$$\displaystyle = \begin{pmatrix} {- 1} & {1} \\ {- 1} & {0} \end{pmatrix} \begin{pmatrix} {0} & {1} \\ {- 1} & {1} \end{pmatrix}$$\displaystyle = \begin{pmatrix} {- 1} & {0} \\ {0} & {- 1} \end{pmatrix}$$= - E$ $2013 = 3 \times 671$이므로 $A ^{2013} = ( - E ) ^{671} = - E$ ∴ $\displaystyle A ^{2013} \begin{pmatrix} {a} \\ {b} \end{pmatrix} = - E \begin{pmatrix} {a} \\ {b} \end{pmatrix} = \begin{pmatrix} {- a} \\ {- b} \end{pmatrix}$ 따라서 $\displaystyle \begin{pmatrix} {- a} \\ {- b} \end{pmatrix} = \begin{pmatrix} {1} \\ {2} \end{pmatrix}$에서 $a = - 1$, $b = - 2$ ∴ $a + b = - 3$