単位行列
\(A({\small\,=\,})\left[~\begin{array}{cccc} \hline a{\tiny 11} & a{\tiny 12} & a{\tiny 13} & a{\tiny 14} \\ \hline a{\tiny 21} & a{\tiny 22} & a{\tiny 23} & a{\tiny 24} \\ \hline a{\tiny 31} & a{\tiny 32} & a{\tiny 33} & a{\tiny 34} \\ \hline a{\tiny 41} & a{\tiny 42} & a{\tiny 43} & a{\tiny 44} \\ \hline \end{array}~\right]\) \(\times\) \(E(=) \left[~\begin{array}{|c|c|c|c|}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}~\right] ~=~ A({\small\,=\,})\left[\begin{array}{cccc} a{\tiny 11} & a{\tiny 12} & a{\tiny 13} & a{\tiny 14} \\ a{\tiny 21} & a{\tiny 22} & a{\tiny 23} & a{\tiny 24} \\ a{\tiny 31} & a{\tiny 32} & a{\tiny 33} & a{\tiny 34} \\ a{\tiny 41} & a{\tiny 42} & a{\tiny 43} & a{\tiny 44} \\ \end{array} \right] \)
\(\begin{array}{l}\textsf{この実数倍$1$の性質をもつ行列$E$を 単位行列 という。単位行列は慣習で$E(\small{\textsf{または}\,I})$ と表記する。} \\ \hspace{10pt}\textsf{行列の基本変形は、単位行列の変形を元に計算していく。}\end{array}\)
\(E(=) \left[~\begin{array}{cccc} \hline 1 & 0 & 0 & 0 \\ \hline 0 & 1 & 0 & 0 \\ \hline 0 & 0 & 1 & 0 \\ \hline 0 & 0 & 0 & 1 \\ \hline \end{array}~\right] \) \(\times\) \(A(=)\left[~\begin{array}{|c|c|c|c|} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \\ \end{array}~\right]\) \(~=~ A(=)\left[\begin{array}{cccc} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \\ \end{array} \right] \)
\(A \times E=A\hspace{15pt}E \times A=A\) すなわち、単位行列はすべての行列との積に可換である。
逆行列とその解法
\(\begin{array}{l}\textsf{逆行列(表記は$A^{\tiny -\,1}$) とは、その元行列 $A$ によって変換された線形空間を双方の行列の積 $A\times A^{\tiny -1}$ により変換前の
線形空間にリセットする行列である。}\end{array}\)
\(\begin{array}{l}\class{Boldfont}{\hspace{240pt}\textsf{逆行列の定義} }\\[10pt]\hspace{20pt}\textsf{正方行列$(\textsf{行要素と列要素が同数である行列})A$に対して、} \\[5pt]\hspace{200pt}\textsf{$AA^{\scriptsize -1}{\small\,=\,}A^{\scriptsize -1}A{\small\,=\,}E$} \\[5pt]\hspace{250pt}\textsf{が成り立つ$A^{\scriptsize -1}(\small\textsf{文字の上添字に$-1$を付けて逆行列を表す})$を $A$の逆行列 という。} \end{array}\)
\[\class{Boldfont}{\textsf{$2\times2$の逆行列の公式}}\]
\(A=\begin{bmatrix}a{\tiny 11}&a{\tiny 12} \\ a{\tiny 21}&a{\tiny 22}\end{bmatrix},\hspace{10pt}A^{\tiny -\,1}={\displaystyle\frac{1}
{a{\tiny 11}a{\tiny 22}{\,\scriptsize -\,}a{\tiny 12}a{\tiny 21}}}{\color{lightgray}\times}\begin{bmatrix}\hspace{5pt}a{\tiny 22}&{\scriptsize -\,}a{\tiny 12} \\ {\scriptsize -\,}a{\tiny 21}
&\hspace{5pt}a{\tiny 11}\end{bmatrix}\)
\[\class{Boldfont}{\textsf{$3\times3$の逆行列の公式}}\]
\(\hspace{-10pt}A=\begin{bmatrix}a{\tiny 11}&a{\tiny 12}&a{\tiny 13} \\ a{\tiny 21}&a{\tiny 22}&a{\tiny 23} \\ a{\tiny 31}&a{\tiny 32}&a{\tiny 33}
\end{bmatrix},\hspace{10pt}A^{\tiny -\,1}={\displaystyle\frac{1}{a{\tiny 11}a{\tiny 22}a{\tiny 33}{\scriptsize +}a{\tiny 12}a{\tiny 23}a{\tiny 31}{\scriptsize +}a{\tiny 13}a{\tiny 21}
a{\tiny 32}{\scriptsize -}a{\tiny 13}a{\tiny 22}a{\tiny 31}{\scriptsize -}a{\tiny 12}a{\tiny 21}a{\tiny 33}{\scriptsize -}a{\tiny 11}a{\tiny 23}a{\tiny 32}}}{\color{lightgray}\times}
\begin{bmatrix}\hspace{5pt}a{\tiny 22}a{\tiny 33}{\scriptsize -}a{\tiny 23}a{\tiny 32}&{\scriptsize -\,}(a{\tiny 12}a{\tiny 33}{\scriptsize -}a{\tiny 13}a{\tiny 32})&a{\tiny 12}a{\tiny 23}{\scriptsize -}a{\tiny 13}a{\tiny 22}
\\ {\scriptsize -\,}(a{\tiny 21}a{\tiny 33}{\scriptsize -}a{\tiny 23}a{\tiny 31})&a{\tiny 11}a{\tiny 33}{\scriptsize -}a{\tiny 13}a{\tiny 31}&{\scriptsize -\,}(a{\tiny 11}a{\tiny 23}{\scriptsize -}a{\tiny 13}a{\tiny 21}) \\ a{\tiny 21}a{\tiny 32}{\scriptsize -}a{\tiny 22}a{\tiny 31}&{\scriptsize -\,}(a{\tiny 11}a{\tiny 32}{\scriptsize -}a{\tiny 12}a{\tiny 31})&a{\tiny 11}a{\tiny 22}{\scriptsize -}a{\tiny 12}a{\tiny 21}\end{bmatrix}\)
\[\class{Boldfont}{\textsf{ガウスの消去法による逆行列の導出}}\]
\(\textsf{行$\href{https://showanojoe.com/template-math/linear-algebra/matrix-6transformation/}{\textsf{基本変形}}$で、実数を用いた行列$A$の逆行列を $\color{red}\underline{\color{black}\textsf{ガウスの消去法}}$ で求める。}\)
\(\color{gray}{\textsf{[事例]}}\)
\(A=\left[\begin{array}{ccc}3&-1&0 \\-1&2&1 \\0&-1&3 \end{array}\right]\)
\(\hspace{-21pt}A|E=\left[\begin{array}{cccc|cccc}{\color{gray}(1)}&3&-1&0&~&1&0&0 \\ {\color{gray}(2)}&-1&2&1&~&0&1&0 \\ {\color{gray}(3)}&0&-1&3&~&0&0&1\end{array}\right]\hspace{10pt}
\small\textsf{行列$A$の${\color{gray}(1)}$行目と${\color{gray}(2)}$行目を入れ替える。$\rightarrow $単位行列$E$の${\color{gray}(1)}$行目と${\color{gray}(2)}$行目を入れ替える。}\)
\(\textsf{②}\left[\begin{array}{cccc|cccc}{\color{gray}(2)}&-1&2&1&~&0&1&0 \\ {\color{gray}(1)}&3 {\color{gray}\tiny +(-3)}&-1 {\color{gray}\tiny
+6}&0 {\color{gray}\tiny +3}&~&1 {\color{gray}\tiny{+0}}&0 {\color{gray}\tiny+3}&0 {\color{gray}\tiny{+0}} \\
{\color{gray}(3)}&0&-1&3&~&0&0&1\end{array}\right]\hspace{10pt}\small\textsf{左行列の${\color{gray}(2)}$行目に$3$を掛けて${\color{gray}(1)}$行目に加える。$\rightarrow $
右行列の${\color{gray}(2)}$行目に$3$を掛けて${\color{gray}(1)}$行目に加える。}\)
\(\textsf{③}\left[\begin{array}{cccc|cccc}{\color{gray}(2)}&-1 {\color{gray}\tiny \times(-1)}&2 {\color{gray}\tiny \times(-1)}&1 {\color{gray}\tiny \times(-1)}&~&0
{\color{gray}\tiny \times(-1)}&1 {\color{gray}\tiny \times(-1)}&0 {\color{gray}\tiny \times(-1)} \\ {\color{gray}(1)}&0&5&3&~&1&3&0 \\ {\color{gray}(3)}&0&-1&3&~&0&0&1\end{array}\right]
\hspace{10pt}\small\textsf{左行列の${\color{gray}(2)}$行目に$-1$を掛ける。$\rightarrow $右行列の${\color{gray}(2)}$行目に$-1$を掛ける。}\)
\(\textsf{④}\left[\begin{array}{cccc|cccc}{\color{gray}(2)}&1&-2&-1&~&0&-1&0 \\ {\color{gray}(1)}&0 {\color{gray}\tiny \times\displaystyle\frac{1}{5}}&5 {\color{gray}\tiny
\times\displaystyle\frac{1}{5}}&3 {\color{gray}\tiny \times\displaystyle\frac{1}{5}}&~&1 {\color{gray}\tiny \times\displaystyle\frac{1}{5}}&3 {\color{gray}\tiny \times\displaystyle
\frac{1}{5}}&0 {\color{gray}\tiny \times\displaystyle\frac{1}{5}} \\ {\color{gray}(3)}&0&-1&3&~&0&0&1\end{array}\right]\hspace{10pt}\small\textsf{左行列の${\color{gray}(1)}$行目に$\displaystyle
\frac{1}{5}$を掛ける。$\rightarrow $右行列の${\color{gray}(1)}$行目に$\displaystyle\frac{1}{5}$を掛ける。}\)
\(\textsf{⑤}\left[\begin{array}{cccc|cccc}{\color{gray}(2)}&1 {\color{gray}\tiny + 0}&-2 {\color{gray}\tiny + 2}&-1 {\color{gray}\tiny +\displaystyle\frac{6}{5}}&~&0
{\color{gray}\tiny +\displaystyle\frac{2}{5}}&-1 {\color{gray}\tiny +\displaystyle\frac{6}{5}}&0 {\color{gray}\tiny +0} \\ {\color{gray}(1)}&0&1&\displaystyle\frac{3}{5}&~&\displaystyle
\frac{1}{5}&\displaystyle\frac{3}{5}&0 \\ {\color{gray}(3)}&0&-1&3&~&0&0&1\end{array}\right]\hspace{10pt}\small\textsf{左行列の${\color{gray}(1)}$行目に$2$を掛けて${\color{gray}(2)}$行目に加える。
$\rightarrow $右行列の${\color{gray}(1)}$行目に$2$を掛けて${\color{gray}(2)}$行目に加える}\)
\(\textsf{⑥}\left[\begin{array}{cccc|cccc}{\color{gray}(2)}&1&0&\displaystyle\frac{1}{5}&~&\displaystyle\frac{2}{5}&\displaystyle\frac{1}{5}&0 \\ {\color{gray}(1)}&0&1
&\displaystyle\frac{3}{5}&~&\displaystyle\frac{1}{5}&\displaystyle\frac{3}{5}&0 \\ {\color{gray}(3)}&0 {\color{gray}\tiny +0}&-1 {\color{gray}\tiny +1}&3 {\color{gray}\tiny
+\displaystyle\frac{3}{5}}&~&0 {\color{gray}\tiny +\displaystyle\frac{1}{5}}&0 {\color{gray}\tiny +\displaystyle\frac{3}{5}}&1 {\color{gray}\tiny +0}\end{array}\right]\hspace{10pt}\small\textsf{
左行列の${\color{gray}(3)}$行目に${\color{gray}(1)}$行目を加える。$\rightarrow $右行列の${\color{gray}(3)}$行目に${\color{gray}(1)}$行目を加える。}\)
\(\textsf{⑦}\left[\begin{array}{cccc|cccc}{\color{gray}(2)}&1&0&\displaystyle\frac{1}{5}&~&\displaystyle\frac{2}{5}&\displaystyle\frac{1}{5}&0 \\ {\color{gray}(1)}&0&1&
\displaystyle\frac{3}{5}&~&\displaystyle\frac{1}{5}&\displaystyle
\frac{3}{5}&0 \\ {\color{gray}(3)}&0 {\color{gray}\tiny \times\displaystyle\frac{5}{18}}&0 {\color{gray}\tiny \times\displaystyle\frac{5}{18}}&\displaystyle\frac{18}{5} {\color{gray}
\tiny \times\displaystyle\frac{5}{18}}&~&\displaystyle\frac{1}{5} {\color{gray}\tiny \times\displaystyle\frac{5}{18}}&\displaystyle\frac{3}{5} {\color{gray}\tiny \times\displaystyle
\frac{5}{18}}&1 {\color{gray}\tiny \times\displaystyle\frac{5}{18}}\end{array}\right]\hspace{10pt}\small\textsf{左行列の${\color{gray}(3)}$行目に
$\displaystyle\frac{5}{18}$を掛ける。$\rightarrow $右行列のの${\color{gray}(3)}$行目に$\displaystyle\frac{5}{18}$を掛ける。}\)
\(\textsf{⑧}\left[\begin{array}{cccc|cccc}{\color{gray}(2)}&1 {\color{gray}\tiny +0}&0 {\color{gray}\tiny +0}&\displaystyle\frac{1}{5} {\color{gray}\tiny +(-\displaystyle
\frac{1}{5})}&~&\displaystyle\frac{2}{5} {\color{gray}\tiny +(-\displaystyle\frac{1}{90})}&\displaystyle\frac{1}{5} {\color{gray}\tiny +(-\displaystyle\frac{1}{30})}&0 {\color{gray}\tiny
+(-\displaystyle\frac{1}{18})} \\ {\color{gray}(1)}&0&1&\displaystyle\frac{3}{5}&~&\displaystyle\frac{1}{5}&\displaystyle\frac{3}{5}&0 \\ {\color{gray}(3)}&0&0&1&~&\displaystyle\frac{1}
{18}&\displaystyle\frac{3}{18}&\displaystyle\frac{5}{18}\end{array}\right]\hspace{10pt}\small\textsf{左行列の${\color{gray}(3)}$行目に$-\displaystyle\frac{1}{5}$を掛けて${\color{gray}(2)}$行目に加える。
$\rightarrow $右行列の${\color{gray}(3)}$行目に$-\displaystyle\frac{1}{5}$を掛けて${\color{gray}(2)}$行目に加える。}\)
\(\textsf{⑨}\left[\begin{array}{cccc|cccc}{\color{gray}(2)}&1&0&0&~&\displaystyle\frac{7}{18}&\displaystyle\frac{1}{6}&-\displaystyle\frac{1}{18} \\ {\color{gray}(1)}&0
{\color{gray}\tiny +0}&1 {\color{gray}\tiny +0}&\displaystyle\frac{3}{5} {\color{gray}\tiny +(-\displaystyle\frac{3}{5})}&~&\displaystyle\frac{1}{5} {\color{gray}\tiny +(-\displaystyle
\frac{1}{30})}&\displaystyle\frac{3}{5} {\color{gray}\tiny +(-\displaystyle\frac{1}{10})}&0 {\color{gray}\tiny +(-\displaystyle\frac{1}{6})} \\ {\color{gray}(3)}&0&0&1&~&\displaystyle
\frac{1}{18}&\displaystyle\frac{1}{6}&\displaystyle\frac{5}{18}\end{array}\right]\hspace{10pt}\small\textsf{左行列の${\color{gray}(3)}$行目に$-\displaystyle\frac{3}{5}$を掛けて${\color{gray}(1)}$
行目に加える。$\rightarrow $右行列の${\color{gray}(3)}$行目に$-\displaystyle\frac{3}{5}$を掛けて${\color{gray}(1)}$行目に加える。}\)
\(\hspace{-26pt}E|A^{\tiny -1}=\left[\begin{array}{cccc|cccc}{\color{gray}(2)}&1&0&0&~&\displaystyle\frac{7}{18}&\displaystyle\frac{1}{6}&-\displaystyle\frac{1}{18} \\ {\color{gray}
(1)}&0&1&0&~&\displaystyle\frac{1}{6}&\displaystyle\frac{1}{2}&-\displaystyle\frac{1}{6} \\ {\color{gray}(3)}&0&0&1&~&\displaystyle\frac{1}{18}&\displaystyle\frac{1}{6}&\displaystyle\frac
{5}{18}\end{array}\right]\)
\(\hspace{20pt}\textsf{したがって}\hspace{10pt}(A=)\left[\begin{array}{ccc}3&-1&0\\-1&2&1\\0&-1&3\end{array}\right]\hspace{10pt} \times \hspace{10pt} (A^{\tiny -1}=)\left[\begin{array}{ccc}\displaystyle
\frac{7}{18}&\displaystyle\frac{1}{6}&-\displaystyle\frac{1}{18} \\ \displaystyle\frac{1}{6}&\displaystyle\frac{1}{2}&-\displaystyle\frac{1}{6} \\ \displaystyle\frac{1}{18}&\displaystyle
\frac{1}{6}&\displaystyle\frac{5}{18}\end{array}\right] \hspace{10pt}=\hspace{10pt}(E=)\left[\begin{array}{l}1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}\right]\hspace{10pt}\textsf{となる。}\)
\(\unitip{}\)