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。

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

解答

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