行列の計算例題（行列式）

　　　　　　　　　　


例題：次の行列式を計算せよ。

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

解答

\( \qquad \qquad \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}0 & 2 & 0 & {-}2 \\ {-}2 & {-}1 & {-}2 & 2 \\ 2 & 1 & {-}2 & 0 \\ 2 & {-}1 & 0 & 1\end{array}\,\right|\qquad \begin{array}{l}\text{非零最小元を探す}\\\qquad(1, 1)\end{array} \)

\( \qquad \qquad \qquad = (-1) \times \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}{-}2 & {-}1 & {-}2 & 2 \\ 0 & 2 & 0 & {-}2 \\ 2 & 1 & {-}2 & 0 \\ 2 & {-}1 & 0 & 1\end{array}\,\right| \qquad \begin{array}{l}\text{第1行と第2行を}\\\quad \text{入れ替える}\end{array} \)

\( \qquad \qquad \qquad = 1 \times \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}{-}1 & {-}2 & {-}2 & 2 \\ 2 & 0 & 0 & {-}2 \\ 1 & 2 & {-}2 & 0 \\ {-}1 & 2 & 0 & 1\end{array}\,\right| \qquad \begin{array}{l}\text{第1列と第2列を}\\\quad\text{入れ替える}\end{array} \)

\( \qquad \qquad \qquad = 1 \times \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}{-}1 & {-}2 & {-}2 & 2 \\ 0 & {-}4 & {-}4 & 2 \\ 0 & 0 & {-}4 & 2 \\ 0 & 4 & 2 & {-}1\end{array}\,\right|\qquad \begin{array}{l}\text{\((1,1)\)成分を使って}\\\text{第１列の成分を小さくする}\\\qquad[0, -2, -1, 1]\end{array} \)

\( \qquad \qquad \qquad = 1 \times (-1) \times \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}4 & {-}4 & 2 \\ 0 & {-}4 & 2 \\ 4 & 2 & {-}1\end{array}\,\right|\qquad\text{段を減らす} \)

\( \qquad \qquad \qquad = (-1) \times \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}4 & {-}4 & 2 \\ 0 & {-}4 & 2 \\ 4 & 2 & {-}1\end{array}\,\right|\qquad \begin{array}{l}\text{非零最小元を探す}\\\qquad(2, 2)\end{array} \)

\( \qquad \qquad \qquad = 1 \times \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}4 & 2 & {-}1 \\ 0 & {-}4 & 2 \\ {-}4 & {-}4 & 2\end{array}\,\right| \qquad \begin{array}{l}\text{第1行と第3行を}\\\quad \text{入れ替える}\end{array} \)

\( \qquad \qquad \qquad = (-1) \times \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & 2 & 4 \\ 2 & {-}4 & 0 \\ 2 & {-}4 & {-}4\end{array}\,\right| \qquad \begin{array}{l}\text{第1列と第3列を}\\\quad\text{入れ替える}\end{array} \)

\( \qquad \qquad \qquad = (-1) \times \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}1 & 2 & 4 \\ 0 & 0 & 8 \\ 0 & 0 & 4\end{array}\,\right|\qquad \begin{array}{l}\text{\((1,1)\)成分を使って}\\\text{第１列の成分を小さくする}\\\qquad[0, -2, -2]\end{array} \)

\( \qquad \qquad \qquad = (-1) \times (-1) \times \left|\,\begin{array}{@{}rwr{15pt}@{}}0 & 8 \\ 0 & 4\end{array}\,\right|\qquad\text{段を減らす} \)

\( \qquad \qquad \qquad = 1 \times \left|\,\begin{array}{@{}rwr{15pt}@{}}0 & 8 \\ 0 & 4\end{array}\,\right| = 1 \times ( 0\times 4 - 8\times 0 ) = 0 \)