行列の計算例題（行列式）

例題：次の行列式を計算せよ。

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

解答

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

\( \qquad \qquad \qquad = (-1) \times \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}wr{15pt}@{}}1 & {-}2 & 0 & {-}2 & 0 \\ 0 & 1 & {-}2 & 1 & {-}1 \\ 2 & 1 & 2 & {-}1 & 1 \\ 2 & 1 & {-}2 & {-}2 & {-}2 \\ 2 & {-}1 & 1 & {-}1 & {-}1\end{array}\,\right| \qquad \begin{array}{l}\text{第1行と第5行を}\\\quad \text{入れ替える}\end{array} \)

\( \qquad \qquad \qquad = (-1) \times \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}wr{15pt}@{}}1 & {-}2 & 0 & {-}2 & 0 \\ 0 & 1 & {-}2 & 1 & {-}1 \\ 0 & 5 & 2 & 3 & 1 \\ 0 & 5 & {-}2 & 2 & {-}2 \\ 0 & 3 & 1 & 3 & {-}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, 0, 2, 2, 2]\end{array} \)

\( \qquad \qquad \qquad = (-1) \times 1 \times \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}1 & {-}2 & 1 & {-}1 \\ 5 & 2 & 3 & 1 \\ 5 & {-}2 & 2 & {-}2 \\ 3 & 1 & 3 & {-}1\end{array}\,\right|\qquad\text{段を減らす} \)

\( \qquad \qquad \qquad = (-1) \times \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}1 & {-}2 & 1 & {-}1 \\ 5 & 2 & 3 & 1 \\ 5 & {-}2 & 2 & {-}2 \\ 3 & 1 & 3 & {-}1\end{array}\,\right|\qquad \begin{array}{l}\text{非零（絶対値）最小元を探す}\\\qquad(0, 0)\end{array} \)

\( \qquad \qquad \qquad = (-1) \times \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}wr{15pt}@{}}1 & {-}2 & 1 & {-}1 \\ 0 & 12 & {-}2 & 6 \\ 0 & 8 & {-}3 & 3 \\ 0 & 7 & 0 & 2\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, 5, 5, 3]\end{array} \)

\( \qquad \qquad \qquad = (-1) \times 1 \times \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}12 & {-}2 & 6 \\ 8 & {-}3 & 3 \\ 7 & 0 & 2\end{array}\,\right|\qquad\text{段を減らす} \)

\( \qquad \qquad \qquad = (-1) \times \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}12 & {-}2 & 6 \\ 8 & {-}3 & 3 \\ 7 & 0 & 2\end{array}\,\right|\qquad \begin{array}{l}\text{非零（絶対値）最小元を探す}\\\qquad(0, 1)\end{array} \)

\( \qquad \qquad \qquad = 1 \times \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}2 & 12 & 6 \\ {-}3 & 8 & 3 \\ 0 & 7 & 2\end{array}\,\right| \qquad \begin{array}{l}\text{第1列と第2列を}\\\quad\text{入れ替える}\end{array} \)

\( \qquad \qquad \qquad = 1 \times \left|\,\begin{array}{@{}rwr{15pt}wr{15pt}@{}}{-}2 & 12 & 6 \\ {-}1 & {-}4 & {-}3 \\ 0 & 7 & 2\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, 1, 0]\end{array} \)

\( \qquad \qquad \qquad = 1 \times \left|\,\begin{array}{@{}rrr@{}}{-}2 & 12 & 6 \\ {-}1 & {-}4 & {-}3 \\ 0 & 7 & 2\end{array}\,\right|\qquad \begin{array}{l}\text{非零（絶対値）最小元を探す}\\\qquad(1, 0)\end{array} \)

\( \qquad \qquad \qquad = (-1) \times \left|\,\begin{array}{@{}rrr@{}}{-}1 & {-}4 & {-}3 \\ {-}2 & 12 & 6 \\ 0 & 7 & 2\end{array}\,\right| \qquad \begin{array}{l}\text{第1行と第2行を}\\\quad \text{入れ替える}\end{array} \)

\( \qquad \qquad \qquad = 1 \times \left|\,\begin{array}{@{}rrr@{}}1 & 4 & 3 \\ {-}2 & 12 & 6 \\ 0 & 7 & 2\end{array}\,\right|\qquad \begin{array}{l}\text{\((1,1)\)成分を}\\\text{monicにする}\end{array} \)

\( \qquad \qquad \qquad = 1 \times \left|\,\begin{array}{@{}rrr@{}}1 & 4 & 3 \\ 0 & 20 & 12 \\ 0 & 7 & 2\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, 0]\end{array} \)

\( \qquad \qquad \qquad = 1 \times 1 \times \left|\,\begin{array}{@{}rr@{}}20 & 12 \\ 7 & 2\end{array}\,\right|\qquad\text{段を減らす} \)

\( \qquad \qquad \qquad = 1 \times \left|\,\begin{array}{@{}rr@{}}20 & 12 \\ 7 & 2\end{array}\,\right| = 1 \times ( 20\times 2 - 12\times 7 ) = -44 \)