행렬 거듭제곱

$\displaystyle A = \begin{pmatrix} - 1 & 3 \\ - 1 & - 1 \end{pmatrix}$에서 $\displaystyle A ^{2} = \begin{pmatrix} - 1 & 3 \\ - 1 & - 1 \end{pmatrix} \begin{pmatrix} - 1 & 3 \\ - 1 & - 1 \end{pmatrix}$$\displaystyle = \begin{pmatrix} - 2 & - 6 \\ 2 & - 2 \end{pmatrix}$ $A ^{3} = AA ^{2}$ $\displaystyle = \begin{pmatrix} - 1 & 3 \\ - 1 & - 1 \end{pmatrix} \begin{pmatrix} - 2 & - 6 \\ 2 & - 2 \end{pmatrix}$ $\displaystyle = \begin{pmatrix} 8 & 0 \\ 0 & 8 \end{pmatrix}$$\displaystyle = 8 \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$$= 8 E$ ∴ $A ^{6} = A ^{3} A ^{3}$ $= 8 ^{2} E ^{2}$ $= 64 E$ ∴ $\displaystyle A ^{6} \begin{pmatrix} 1 \\ 1 \end{pmatrix} = 64 E \begin{pmatrix} 1 \\ 1 \end{pmatrix}$$\displaystyle = \begin{pmatrix} 64 \\ 64 \end{pmatrix}$ ∴ $a = 64 , b = 64$ ∴ $a + b = 128$