행렬 성분 조건

$A ^{2}$$\displaystyle = \left( \begin{array}{lll} \begin{array}{rrr} a _{11} & & a _{12} \\ a _{21} & & a _{22} \end{array} \end{array} \right) \left( \begin{array}{lll} \begin{array}{rrr} a _{11} & & a _{12} \\ a _{21} & & a _{22} \end{array} \end{array} \right)$$\displaystyle = \left( \begin{array}{lll} \begin{matrix} a _{11} \mathit{^{2}} + a _{12} a _{21} & & a _{12} \left( a _{11} + a _{22} \right) \\ a _{21} \left( a _{11} + a _{22} \right) & & a _{12} a _{21} + a _{22} \mathit{^{2}} \end{matrix} \end{array} \right)$$\displaystyle = \left( \begin{array}{lll} \begin{array}{rrr} b _{11} & & b _{12} \\ b _{21} & & b _{22} \end{array} \end{array} \right)$ 조건 (가)에서 $i = 1$, $j = 2$일 때, $a _{12} \mathit{^{2}} \left( a _{11} + a _{22} \right) = 0$ $a _{12} = 0$이면 $\displaystyle A = \left( \begin{array}{lll} \begin{array}{rrr} a _{11} & & 0 \\ a _{21} & & a _{22} \end{array} \end{array} \right)$, $\displaystyle B = \left( \begin{array}{lll} \begin{matrix} a _{11} \mathit{^{2}} & & 0 \\ a _{21} \left( a _{11} + a _{22} \right) & & a _{22} \mathit{^{2}} \end{matrix} \end{array} \right)$ $a _{12} + b _{12} = 0$이므로 조건 (나)를 만족시키지 않는다. 그러므로 $a _{12} \neq 0$이고 $a _{11} + a _{22} = 0$ $\displaystyle A = \left( \begin{array}{lll} \begin{array}{rrr} a _{11} & & a _{12} \\ a _{21} & & - a _{11} \end{array} \end{array} \right)$, $\displaystyle B = \left( \begin{array}{lll} \begin{matrix} a _{11} \mathit{^{2}} + a _{12} a _{21} & & 0 \\ 0 & & a _{11} \mathit{^{2}} + a _{12} a _{21} \end{matrix} \end{array} \right)$ 조건 (가)에서 $i = 1$, $j = 1$에 대하여 $a _{11} \times \left( a _{11} \mathit{^{2}} + a _{12} a _{21} \right) = 0$ 행렬 $B$가 영행렬이 아니므로, $a _{11} \mathit{^{2}} + a _{12} a _{21} \ne 0$이다. 그러므로 $a _{11} = 0$ 즉, 두 행렬 $A$, $B$는 $\displaystyle A = \left( \begin{array}{lll} \begin{array}{rrr} 0 & & a _{12} \\ a _{21} & & 0 \end{array} \end{array} \right)$, $\displaystyle B = \left( \begin{array}{lll} \begin{array}{rrr} a _{12} a _{21} & & 0 \\ 0 & & a _{12} a _{21} \end{array} \end{array} \right)$이고, $\displaystyle A + B = \left( \begin{array}{lll} \begin{array}{rrr} a _{12} a _{21} & & a _{12} \\ a _{21} & & a _{12} a _{21} \end{array} \end{array} \right)$의 모든 성분의 곱이 $- 8$이므로 $a _{12} \mathit{^{3}} a _{21} \mathit{^{3}} = - 8$이고, $a _{12} a _{21}$이 실수이므로 $a _{12} a _{21} = - 2$ $\cdots$$\cdots$ ㉠ $A + B$의 모든 성분의 합이 $- 1$이므로 $a _{12} + a _{21} + 2 a _{12} a _{21} = - 1$ $a _{12} + a _{21} = 3$ $\cdots$$\cdots$ ㉡ ㉠, ㉡에서 $a _{12} \mathit{^{3}} + a _{21} \mathit{^{3}}$$= \left( a _{12} + a _{21} \right) ^{3} - 3 a _{12} a _{21} \left( a _{12} + a _{21} \right)$ $= 3 ^{3} - 3 \times \left( - 2 \right) \times 3 = 27 + 18 = 45$