행렬과 역행렬 조건

[출제의도] 행렬의 연산을 활용하여 문제해결하기 $\displaystyle A = {\begin{pmatrix} a & b \\ c & d \end{pmatrix}}$라 할 때 (가)에서 $\displaystyle {\begin{pmatrix} a & b \\ c & d \end{pmatrix}} {\begin{pmatrix} 1 \\ 2 \end{pmatrix}} = {\begin{pmatrix} 3 \\ 4 \end{pmatrix}}$ $\cdots \cdots$㉠ (나)의 양변에 행렬 $A$를 곱하면 $\displaystyle A ^{2} \left( \begin{array}{l} \begin{matrix} 2 \\ 4 \end{matrix} \end{array} \right) + E \left( \begin{array}{l} \begin{matrix} 1 \\ 0 \end{matrix} \end{array} \right) = A \left( \begin{array}{l} \begin{matrix} 1 \\ 2 \end{matrix} \end{array} \right)$ (단, $E$는 단위행렬이다.) $\displaystyle 2 AA \left( \begin{array}{l} \begin{matrix} 1 \\ 2 \end{matrix} \end{array} \right) + E \left( \begin{array}{l} \begin{matrix} 1 \\ 0 \end{matrix} \end{array} \right) = A \left( \begin{array}{l} \begin{matrix} 1 \\ 2 \end{matrix} \end{array} \right)$ $\displaystyle 2 A \left( \begin{array}{l} \begin{matrix} 3 \\ 4 \end{matrix} \end{array} \right) + E \left( \begin{array}{l} \begin{matrix} 1 \\ 0 \end{matrix} \end{array} \right) = \left( \begin{array}{l} \begin{matrix} 3 \\ 4 \end{matrix} \end{array} \right)$ $\displaystyle 2 A \left( \begin{array}{l} \begin{matrix} 3 \\ 4 \end{matrix} \end{array} \right) = \left( \begin{array}{l} \begin{matrix} 3 \\ 4 \end{matrix} \end{array} \right) - \left( \begin{array}{l} \begin{matrix} 1 \\ 0 \end{matrix} \end{array} \right) = \left( \begin{array}{l} \begin{matrix} 2 \\ 4 \end{matrix} \end{array} \right)$ $\displaystyle A \left( \begin{array}{l} \begin{matrix} 3 \\ 4 \end{matrix} \end{array} \right) = \left( \begin{array}{l} \begin{matrix} 1 \\ 2 \end{matrix} \end{array} \right)$ $\displaystyle {\begin{pmatrix} a & b \\ c & d \end{pmatrix}} {\begin{pmatrix} 3 \\ 4 \end{pmatrix}} = {\begin{pmatrix} 1 \\ 2 \end{pmatrix}}$ $\cdots \cdots$㉡ ㉠, ㉡에서 $\displaystyle {\begin{cases} a + 2 b = 3 _{{} _{{} _{}}} \\ c + 2 d = 4 \end{cases}}$ , $\displaystyle {\begin{cases} 3 a + 4 b = 1 _{{} _{{} _{}}} \\ 3 c + 4 d = 2 \end{cases}}$ $\therefore$ $\displaystyle A = {\begin{pmatrix} - 5 & 4 \\ - 6 & 5 \end{pmatrix}}$ 따라서 행렬 $A$의 모든 성분의 합은 $- 2$