역행렬 조건 행렬

조건 (가)에 의하여 $( A - E ) ( A - 3 E ) = E$ $A ^{2} = 4 A - 2 E$ ⋯⋯ ㉠ 조건 (나)에서 $\displaystyle A \begin{pmatrix} {- 1} \\ {2} \end{pmatrix} = \begin{pmatrix} {2} \\ {0} \end{pmatrix}$이므로 $\displaystyle A \begin{pmatrix} {2} \\ {0} \end{pmatrix}$$\displaystyle = A ^{2} \begin{pmatrix} {- 1} \\ {2} \end{pmatrix}$$\displaystyle = ( 4 A - 2 E ) \begin{pmatrix} {- 1} \\ {2} \end{pmatrix}$$\displaystyle = 4 A \begin{pmatrix} {- 1} \\ {2} \end{pmatrix} - 2 E \begin{pmatrix} {- 1} \\ {2} \end{pmatrix}$ $\displaystyle = 4 \begin{pmatrix} {2} \\ {0} \end{pmatrix} - 2 \begin{pmatrix} {- 1} \\ {2} \end{pmatrix}$$\displaystyle = \begin{pmatrix} {10} \\ {- 4} \end{pmatrix}$ ∴ $\displaystyle A \begin{pmatrix} {1} \\ {0} \end{pmatrix}$$\displaystyle = \begin{pmatrix} {5} \\ {- 2} \end{pmatrix}$ ∴ $\displaystyle A \begin{pmatrix} {3} \\ {0} \end{pmatrix}$$\displaystyle = 3 A \begin{pmatrix} {1} \\ {0} \end{pmatrix}$$\displaystyle = 3 \begin{pmatrix} {5} \\ {- 2} \end{pmatrix}$$\displaystyle = \begin{pmatrix} {15} \\ {- 6} \end{pmatrix}$ ∴ $x = 15$, $y = - 6$ ∴ $x + y = 9$