행렬 거듭제곱과 역행렬

$\displaystyle A ^{2} = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$ $\displaystyle A ^{3} = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 3 \\ 0 & 1 \end{pmatrix}$ ⋮ $\displaystyle A ^{10} = \begin{pmatrix} 1 & 10 \\ 0 & 1 \end{pmatrix}$ $\displaystyle ( A ^{- 1} ) ^{10} = ( A ^{10} ) ^{- 1} = \begin{pmatrix} 1 & - 10 \\ 0 & 1 \end{pmatrix}$ ∴ $\displaystyle A ^{10} + ( A ^{- 1} ) ^{10} = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}$ 따라서 모든 성분의 합은 $4$이다.