행렬 조건 성분 합

(가)에서 $\displaystyle A \left( \begin{array}{l} \begin{matrix} 2 \\ 3 \end{matrix} \end{array} \right) + A ^{- 1} \left( \begin{array}{l} \begin{matrix} 2 \\ 3 \end{matrix} \end{array} \right) = E \left( \begin{array}{l} \begin{matrix} 2 \\ 3 \end{matrix} \end{array} \right)$ (나)에서 $\displaystyle \left( \begin{array}{l} \begin{matrix} 3 \\ 5 \end{matrix} \end{array} \right) + A ^{- 1} \left( \begin{array}{l} \begin{matrix} 2 \\ 3 \end{matrix} \end{array} \right) = \left( \begin{array}{l} \begin{matrix} 2 \\ 3 \end{matrix} \end{array} \right)$ $\displaystyle A ^{- 1} \left( \begin{array}{l} \begin{matrix} 2 \\ 3 \end{matrix} \end{array} \right) = \left( \begin{array}{l} \begin{matrix} 2 \\ 3 \end{matrix} \end{array} \right) - \left( \begin{array}{l} \begin{matrix} 3 \\ 5 \end{matrix} \end{array} \right) = \left( \begin{array}{l} \begin{matrix} - 1 \\ - 2 \end{matrix} \end{array} \right)$ $\therefore$ $\displaystyle A \left( \begin{array}{l} \begin{matrix} - 1 \\ - 2 \end{matrix} \end{array} \right) = \left( \begin{array}{l} \begin{matrix} 2 \\ 3 \end{matrix} \end{array} \right)$ $\displaystyle A = {\begin{pmatrix} a & b \\ c & d \end{pmatrix}}$로 놓으면 $\displaystyle {\begin{pmatrix} a & b \\ c & d \end{pmatrix}} \left( \begin{array}{l} \begin{matrix} 2 \\ 3 \end{matrix} \end{array} \right) = \left( \begin{array}{l} \begin{matrix} 3 \\ 5 \end{matrix} \end{array} \right)$, $\displaystyle {\begin{pmatrix} a & b \\ c & d \end{pmatrix}} \left( \begin{array}{l} \begin{matrix} - 1 \\ - 2 \end{matrix} \end{array} \right) = \left( \begin{array}{l} \begin{matrix} 2 \\ 3 \end{matrix} \end{array} \right)$이므로 $2 a + 3 b = 3 , 2 c + 3 d = 5$ $\cdots$㉠ $- a - 2 b = 2 , - c - 2 d = 3$ $\cdots$㉡ ㉠, ㉡을 연립하여 풀면 $a = 12 , b = - 7 , c = 19 , d = - 11$ $\therefore$ $A =$$\displaystyle {\begin{pmatrix} 12 & - 7 \\ 19 & - 11 \end{pmatrix}}$ 따라서 행렬 $A$의 모든 성분의 합은 $13$이다. [다른풀이1] (가)에서 $\displaystyle A \left( \begin{array}{l} \begin{matrix} 2 \\ 3 \end{matrix} \end{array} \right) + A ^{- 1} \left( \begin{array}{l} \begin{matrix} 2 \\ 3 \end{matrix} \end{array} \right) = E \left( \begin{array}{l} \begin{matrix} 2 \\ 3 \end{matrix} \end{array} \right)$ (나)의 $\displaystyle A \left( \begin{array}{l} \begin{matrix} 2 \\ 3 \end{matrix} \end{array} \right) = \left( \begin{array}{l} \begin{matrix} 3 \\ 5 \end{matrix} \end{array} \right)$ … ㉠을 대입하면 $\displaystyle \left( \begin{array}{l} \begin{matrix} 3 \\ 5 \end{matrix} \end{array} \right) + A ^{- 1} \left( \begin{array}{l} \begin{matrix} 2 \\ 3 \end{matrix} \end{array} \right) = \left( \begin{array}{l} \begin{matrix} 2 \\ 3 \end{matrix} \end{array} \right)$에서 $\displaystyle A ^{- 1} \left( \begin{array}{l} \begin{matrix} 2 \\ 3 \end{matrix} \end{array} \right) = \left( \begin{array}{l} \begin{matrix} 2 \\ 3 \end{matrix} \end{array} \right) - \left( \begin{array}{l} \begin{matrix} 3 \\ 5 \end{matrix} \end{array} \right) = \left( \begin{array}{l} \begin{matrix} - 1 \\ - 2 \end{matrix} \end{array} \right)$ $\therefore$ $\displaystyle A \left( \begin{array}{l} \begin{matrix} - 1 \\ - 2 \end{matrix} \end{array} \right) = \left( \begin{array}{l} \begin{matrix} 2 \\ 3 \end{matrix} \end{array} \right)$ … ㉡ ㉠, ㉡에서 $\displaystyle A {\begin{pmatrix} 2 & - 1 \\ 3 & - 2 \end{pmatrix}} = {\begin{pmatrix} 3 & 2 \\ 5 & 3 \end{pmatrix}}$ $\displaystyle A = {\begin{pmatrix} 3 & 2 \\ 5 & 3 \end{pmatrix}} {\begin{pmatrix} 2 & - 1 \\ 3 & - 2 \end{pmatrix}} ^{- 1}$ $\displaystyle = {\begin{pmatrix} 3 & 2 \\ 5 & 3 \end{pmatrix}} {\begin{pmatrix} 2 & - 1 \\ 3 & - 2 \end{pmatrix}}$$\displaystyle = {\begin{pmatrix} 12 & - 7 \\ 19 & - 11 \end{pmatrix}}$ 따라서 행렬 $A$의 모든 성분의 합은 $13$이다. [다른풀이2] (가)의 양변에 행렬 $A$를 곱하면 $A ^{2} + E = A$ $\displaystyle A ^{2} \left( \begin{array}{l} \begin{matrix} 2 \\ 3 \end{matrix} \end{array} \right) + E \left( \begin{array}{l} \begin{matrix} 2 \\ 3 \end{matrix} \end{array} \right) = A \left( \begin{array}{l} \begin{matrix} 2 \\ 3 \end{matrix} \end{array} \right)$에서 (나)의 $\displaystyle A \left( \begin{array}{l} \begin{matrix} 2 \\ 3 \end{matrix} \end{array} \right) = \left( \begin{array}{l} \begin{matrix} 3 \\ 5 \end{matrix} \end{array} \right)$ … ㉠을 대입하면 $\displaystyle A \left( \begin{array}{l} \begin{matrix} 3 \\ 5 \end{matrix} \end{array} \right) + \left( \begin{array}{l} \begin{matrix} 2 \\ 3 \end{matrix} \end{array} \right) = \left( \begin{array}{l} \begin{matrix} 3 \\ 5 \end{matrix} \end{array} \right)$ $\therefore$ $\displaystyle A \left( \begin{array}{l} \begin{matrix} 3 \\ 5 \end{matrix} \end{array} \right) = \left( \begin{array}{l} \begin{matrix} 1 \\ 2 \end{matrix} \end{array} \right)$ … ㉡ ㉠, ㉡에서 $\displaystyle A {\begin{pmatrix} 2 & 3 \\ 3 & 5 \end{pmatrix}} = {\begin{pmatrix} 3 & 1 \\ 5 & 2 \end{pmatrix}}$ $\displaystyle A = {\begin{pmatrix} 3 & 1 \\ 5 & 2 \end{pmatrix}} {\begin{pmatrix} 2 & 3 \\ 3 & 5 \end{pmatrix}} ^{- 1}$$\displaystyle = {\begin{pmatrix} 3 & 1 \\ 5 & 2 \end{pmatrix}} {\begin{pmatrix} 5 & - 3 \\ - 3 & 2 \end{pmatrix}}$$\displaystyle = {\begin{pmatrix} 12 & - 7 \\ 19 & - 11 \end{pmatrix}}$ 따라서 행렬 $A$의 모든 성분의 합은 $13$이다.