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。

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

解答

係数行列を簡約化する。\[ \left[\begin{array}{@{}rwr{20pt}wr{20pt}wr{20pt}wr{20pt}wr{20pt}@{}}{-}5 & {-}2 & {-}3 & 2 & {-}7 & 6 \\ 3 & 1 & 2 & {-}1 & 3 & {-}1 \\ {-}2 & {-}1 & {-}1 & 1 & {-}4 & 5 \\ 1 & 0 & 1 & 0 & {-}1 & 4 \\ {-}3 & {-}1 & {-}2 & 1 & {-}3 & 1\end{array}\right]\ \to\ \left[\begin{array}{@{}rwr{20pt}wr{20pt}wr{20pt}wr{20pt}wr{20pt}@{}}1 & 0 & 1 & 0 & {-}1 & 4 \\ 0 & 1 & {-}1 & {-}1 & 6 & {-}13 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0\end{array}\right] \] \[ \begin{array}{l|cccccc} 主成分を含む列 & 1&2\\\hline 主成分を含まない列 & 3&4&5&6\\\hline 主成分以外の成分 & \begin{array}{@{}r@{}}1 \\ {-}1\end{array}&\begin{array}{@{}r@{}}0 \\ {-}1\end{array}&\begin{array}{@{}r@{}}{-}1 \\ 6\end{array}&\begin{array}{@{}r@{}}4 \\ {-}13\end{array}\end{array} \] 従って、解は \[ x=k_0\left[\begin{array}{@{}r@{}}1 \\ {-}1 \\ {-}1 \\ 0 \\ 0 \\ 0\end{array}\right]+k_1\left[\begin{array}{@{}r@{}}0 \\ {-}1 \\ 0 \\ {-}1 \\ 0 \\ 0\end{array}\right]+k_2\left[\begin{array}{@{}r@{}}{-}1 \\ 6 \\ 0 \\ 0 \\ {-}1 \\ 0\end{array}\right]+k_3\left[\begin{array}{@{}r@{}}4 \\ {-}13 \\ 0 \\ 0 \\ 0 \\ {-}1\end{array}\right] \]