행렬 곱 조건

$\displaystyle B = \left( \begin{array}{lll} \begin{array}{rrr} b _{11} & & b _{12} \\ b _{21} & & b _{22} \end{array} \end{array} \right)$, $\displaystyle C = \left( \begin{array}{lll} \begin{array}{rrr} c _{11} & & c _{12} \\ c _{21} & & c _{22} \end{array} \end{array} \right)$라 하자. 조건 (가)에서 $A B$$\displaystyle = \left( \begin{array}{lll} \begin{array}{rrr} 0 & & 0 \\ 6 & & 0 \end{array} \end{array} \right) \left( \begin{array}{lll} \begin{array}{rrr} b _{11} & & b _{12} \\ b _{21} & & b _{22} \end{array} \end{array} \right)$$\displaystyle = \left( \begin{array}{lll} \begin{array}{rrr} 0 & & 0 \\ 6 b _{11} & & 6 b _{12} \end{array} \end{array} \right) = O$ 이므로 $b _{11} = b _{12} = 0$이고, $C A$$\displaystyle = \left( \begin{array}{lll} \begin{array}{rrr} c _{11} & & c _{12} \\ c _{21} & & c _{22} \end{array} \end{array} \right) \left( \begin{array}{lll} \begin{array}{rrr} 0 & & 0 \\ 6 & & 0 \end{array} \end{array} \right)$$\displaystyle = \left( \begin{array}{lll} \begin{array}{rrr} 6 c _{12} & & 0 \\ 6 c _{22} & & 0 \end{array} \end{array} \right) = O$ 이므로 $c _{12} = c _{22} = 0$이다. 조건 (나)에서 $b _{21} + b _{22} = 3$, $c _{11} = c _{21}$ $B C = A$이므로 $\displaystyle \left( \begin{array}{lll} \begin{matrix} 0 & & 0 \\ b _{21} & & 3 - b _{21} \end{matrix} \end{array} \right) \left( \begin{array}{lll} \begin{array}{rrr} c _{11} & & 0 \\ c _{11} & & 0 \end{array} \end{array} \right) = \left( \begin{array}{lll} \begin{array}{rrr} 0 & & 0 \\ 3 c _{11} & & 0 \end{array} \end{array} \right) = \left( \begin{array}{lll} \begin{array}{rrr} 0 & & 0 \\ 6 & & 0 \end{array} \end{array} \right)$ 에서 $c _{11} = 2$ 그러므로 $\displaystyle C = \left( \begin{array}{lll} \begin{array}{rrr} 2 & & 0 \\ 2 & & 0 \end{array} \end{array} \right)$ 따라서 행렬 $C$의 모든 성분의 합은 $4$