행렬 조건과 벡터 변환

[출제의도] 행렬의 연산과 역행렬의 성질을 이용하여 조건을 만족시키는 행렬을 구한다. 조건 (가)에서 $2 A ^{2} - A = E$ $A ( 2 A - E ) = E$ $A ^{- 1} = 2 A - E$ … ㉠ 조건 (나)에서 등식의 양변의 왼쪽에 $A ^{- 1}$을 곱하면 $\displaystyle \left( \begin{array}{l} {\begin{matrix} 3 \\ 0 \end{matrix}} \end{array} \right)$$= A ^{- 1}$$\displaystyle \left( \begin{array}{l} {\begin{matrix} 1 \\ 2 \end{matrix}} \end{array} \right)$ ㉠에서 $\displaystyle \left( \begin{array}{l} {\begin{matrix} 3 \\ 0 \end{matrix}} \end{array} \right)$$= ( 2 A - E )$$\displaystyle \left( \begin{array}{l} {\begin{matrix} 1 \\ 2 \end{matrix}} \end{array} \right)$ $\displaystyle 2 A \left( \begin{array}{l} {\begin{matrix} 1 \\ 2 \end{matrix}} \end{array} \right)$$-$$\displaystyle \left( \begin{array}{l} {\begin{matrix} 1 \\ 2 \end{matrix}} \end{array} \right)$$=$$\displaystyle \left( \begin{array}{l} {\begin{matrix} 3 \\ 0 \end{matrix}} \end{array} \right)$ $\displaystyle 2 A \left( \begin{array}{l} {\begin{matrix} 1 \\ 2 \end{matrix}} \end{array} \right)$$=$$\displaystyle \left( \begin{array}{l} {\begin{matrix} 4 \\ 2 \end{matrix}} \end{array} \right)$ $\displaystyle A \left( \begin{array}{l} {\begin{matrix} 1 \\ 2 \end{matrix}} \end{array} \right)$$=$$\displaystyle \left( \begin{array}{l} {\begin{matrix} 2 \\ 1 \end{matrix}} \end{array} \right)$ … ㉡ 조건 (다)에서 등식의 양변의 왼쪽에 $B$를 곱하면 $B ( A - B ^{- 1} )$$\displaystyle \left( \begin{array}{l} {\begin{matrix} 1 \\ 2 \end{matrix}} \end{array} \right)$$=$$B$$\displaystyle \left( \begin{array}{l} {\begin{matrix} 0 \\ 0 \end{matrix}} \end{array} \right)$ $BA$$\displaystyle \left( \begin{array}{l} {\begin{matrix} 1 \\ 2 \end{matrix}} \end{array} \right)$$-$$\displaystyle \left( \begin{array}{l} {\begin{matrix} 1 \\ 2 \end{matrix}} \end{array} \right)$$=$$\displaystyle \left( \begin{array}{l} {\begin{matrix} 0 \\ 0 \end{matrix}} \end{array} \right)$ $BA$$\displaystyle \left( \begin{array}{l} {\begin{matrix} 1 \\ 2 \end{matrix}} \end{array} \right)$$=$$\displaystyle \left( \begin{array}{l} {\begin{matrix} 1 \\ 2 \end{matrix}} \end{array} \right)$ ㉡에서 $BA$$\displaystyle \left( \begin{array}{l} {\begin{matrix} 1 \\ 2 \end{matrix}} \end{array} \right)$$= B$$\displaystyle \left( \begin{array}{l} {\begin{matrix} 2 \\ 1 \end{matrix}} \end{array} \right)$$=$$\displaystyle \left( \begin{array}{l} {\begin{matrix} 1 \\ 2 \end{matrix}} \end{array} \right)$이므로 $B$$\displaystyle \left( \begin{array}{l} {\begin{matrix} 2 \\ 1 \end{matrix}} \end{array} \right)$$=$$\displaystyle \left( \begin{array}{l} {\begin{matrix} 1 \\ 2 \end{matrix}} \end{array} \right)$ ∴ $a = 1$, $b = 2$이므로 $a + b = 3$ [다른 풀이] 조건 (가)에서 등식의 양변의 오른쪽에 행렬 $\displaystyle \left( \begin{array}{l} {\begin{matrix} 3 \\ 0 \end{matrix}} \end{array} \right)$을 곱하면 $\displaystyle ( 2 A ^{2} - A - E ) \left( \begin{array}{l} {\begin{matrix} 3 \\ 0 \end{matrix}} \end{array} \right) = \left( \begin{array}{l} {\begin{matrix} 0 \\ 0 \end{matrix}} \end{array} \right)$ $\displaystyle 2 A ^{2} {\begin{pmatrix} 3 \\ 0 \end{pmatrix}} - A {\begin{pmatrix} 3 \\ 0 \end{pmatrix}} - {\begin{pmatrix} 3 \\ 0 \end{pmatrix}} = \left( \begin{array}{l} {\begin{matrix} 0 \\ 0 \end{matrix}} \end{array} \right)$ 조건 (나)에서 $\displaystyle 2 A {\begin{pmatrix} 1 \\ 2 \end{pmatrix}} - {\begin{pmatrix} 1 \\ 2 \end{pmatrix}} - {\begin{pmatrix} 3 \\ 0 \end{pmatrix}} = \left( \begin{array}{l} {\begin{matrix} 0 \\ 0 \end{matrix}} \end{array} \right)$ $\displaystyle A {\begin{pmatrix} 1 \\ 2 \end{pmatrix}} = {\begin{pmatrix} 2 \\ 1 \end{pmatrix}}$ … ㉠ 조건 (다)와 ㉠에서 $\displaystyle A \begin{pmatrix} 1 \\ 2 \end{pmatrix} - B ^{- 1} \begin{pmatrix} 1 \\ 2 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}$ $B ^{-1}$$\displaystyle \left( \begin{array}{l} {\begin{matrix} 1 \\ 2 \end{matrix}} \end{array} \right)$$=$$\displaystyle A \begin{pmatrix} 1 \\ 2 \end{pmatrix}$$\displaystyle = \left( \begin{array}{l} {\begin{matrix} 2 \\ 1 \end{matrix}} \end{array} \right)$ 이 등식의 양변의 왼쪽에 $B$를 곱하면 $B$$\displaystyle \left( \begin{array}{l} {\begin{matrix} 2 \\ 1 \end{matrix}} \end{array} \right)$$=$$\displaystyle \left( \begin{array}{l} {\begin{matrix} 1 \\ 2 \end{matrix}} \end{array} \right)$ ∴ $a + b = 3$