行列の計算例題（斉次形連立一次方程式）

例題：次の斉次形連立一次方程式の解を求めよ。

\[ \left[\begin{array}{@{}rwr{20pt}wr{20pt}wr{20pt}wr{20pt}wr{20pt}@{}}{-}2 & {-}5 & 6 & {-}7 & {-}3 & {-}3 \\ 3 & 2 & {-}4 & 2 & {-}1 & 0 \\ 1 & {-}3 & 2 & {-}5 & {-}4 & {-}3 \\ {-}1 & 3 & {-}2 & 5 & 4 & 3 \\ 4 & {-}1 & {-}2 & {-}3 & {-}5 & {-}3\end{array}\right]\,x\,=\,0 \]

解答

係数行列を簡約化する。\[ \left[\begin{array}{@{}rwr{20pt}wr{20pt}wr{20pt}wr{20pt}wr{20pt}@{}}{-}2 & {-}5 & 6 & {-}7 & {-}3 & {-}3 \\ 3 & 2 & {-}4 & 2 & {-}1 & 0 \\ 1 & {-}3 & 2 & {-}5 & {-}4 & {-}3 \\ {-}1 & 3 & {-}2 & 5 & 4 & 3 \\ 4 & {-}1 & {-}2 & {-}3 & {-}5 & {-}3\end{array}\right]\ \to\ \left[\begin{array}{@{}rwr{20pt}wr{20pt}wr{20pt}wr{20pt}wr{20pt}@{}}1 & 0 & {-}8/11 & {-}4/11 & {-}1 & {-}6/11 \\ 0 & 1 & {-}10/11 & 17/11 & 1 & 9/11 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0\end{array}\right] \] \[ \begin{array}{l|cccccc} 主成分を含む列 & 1&2\\\hline 主成分を含まない列 & 3&4&5&6\\\hline 主成分以外の成分 & \begin{array}{@{}r@{}}{-}8/11 \\ {-}10/11\end{array}&\begin{array}{@{}r@{}}{-}4/11 \\ 17/11\end{array}&\begin{array}{@{}r@{}}{-}1 \\ 1\end{array}&\begin{array}{@{}r@{}}{-}6/11 \\ 9/11\end{array}\end{array} \] 従って、解は \[ x=k_0\left[\begin{array}{@{}r@{}}{-}8 \\ {-}10 \\ {-}11 \\ 0 \\ 0 \\ 0\end{array}\right]+k_1\left[\begin{array}{@{}r@{}}{-}4 \\ 17 \\ 0 \\ {-}11 \\ 0 \\ 0\end{array}\right]+k_2\left[\begin{array}{@{}r@{}}{-}1 \\ 1 \\ 0 \\ 0 \\ {-}1 \\ 0\end{array}\right]+k_3\left[\begin{array}{@{}r@{}}{-}6 \\ 9 \\ 0 \\ 0 \\ 0 \\ {-}11\end{array}\right] \]