基本変形と行列式 のバックアップの現在との差分(No.2)

  • 追加された行はこの色です。
  • 削除された行はこの色です。
【[[2]]】数学 Mathematics
【[[2.2]]】代数学 Algebra
【[[2.2.3]]】線形代数学 Linear algebra
【[[2.2.3.3]]】行列式 Determinant
【[[2.2.3.3.3]]】行列式の計算
【[[2>Category.2]]】数学 Mathematics
【[[2.2>Category.2.2]]】代数学 Algebra
【[[2.2.3>Category.2.2.3]]】線形代数学 Linear algebra
【[[2.2.3.3>Category.2.2.3.3]]】行列式 Determinant
【[[2.2.3.3.3>Category.2.2.3.3.3]]】行列式の計算

CENTER:|CENTER:【2.2.3.3.2】 ''[[行列式の計算]]''|
CENTER:|CENTER:【[[2.2.3.3.3>Category.2.2.3.3.3]]】 ''[[行列式の計算]]''|
|【2.2.3.3.3.a】[[基本変形と行列式]]&BR;【2.2.3.3.3.b】[[三角行列の行列式]]&BR;【2.2.3.3.3.c】[[スカラー値を返す関数と行列式]]&BR;【2.2.3.3.3.d】[[転地行列と行列式]]&BR;【2.2.3.3.3.e】[[行列式の分解]]|




※ このページではMathJax を利用しております。環境によっては数式を表示できないかもしれません。


#contents


*【P-1】別の列に実数倍加算の証明 [#v8a0e7aa]
ある列を実数倍して、別の列に足しても、行列式は等しいことを証明する。

&mathjax{\left| \begin{array}{ccccccc} a_{11} & \dots & \textcolor{blue}{a_{1j}} & \dots & \textcolor{red}{a_{1k}}+r\textcolor{blue}{a_{1k}} & \dots & a_{1n} \\ a_{21} & & \textcolor{blue}{a_{2j}} & & \textcolor{red}{a_{2k}}+r\textcolor{blue}{a_{2k}} & & a_{2n} \\ \vdots & & \vdots & & \vdots & & \vdots \\a_{n1} & \dots & \textcolor{blue}{a_{nj}} & \dots & \textcolor{red}{a_{nk}}+r\textcolor{blue}{a_{nk}} & \dots & a_{nn} \end{array}\right|};

&color(Maroon){線形多重性より、};


&mathjax{=\left| \begin{array}{ccccccc} a_{11} & \dots & \textcolor{blue}{a_{1j}} & \dots & \textcolor{red}{a_{1k}} & \dots & a_{1n} \\ a_{21} & & \textcolor{blue}{a_{2j}} & & \textcolor{red}{a_{2k}} & & a_{2n} \\ \vdots & & \vdots & & \vdots & & \vdots \\a_{n1} & \dots & \textcolor{blue}{a_{nj}} & \dots & \textcolor{red}{a_{nk}} & \dots & a_{nn} \end{array}\right|};&mathjax{+ \Large r \normalsize \left| \begin{array}{ccccccc} a_{11} & \dots & \textcolor{blue}{a_{1j}} & \dots & \textcolor{blue}{a_{1k}} & \dots & a_{1n} \\ a_{21} & & \textcolor{blue}{a_{2j}} & & \textcolor{blue}{a_{2k}} & & a_{2n} \\ \vdots & & \vdots & & \vdots & & \vdots \\a_{n1} & \dots & \textcolor{blue}{a_{nj}} & \dots & \textcolor{blue}{a_{nk}} & \dots & a_{nn} \end{array}\right|};

&color(Maroon){同じ列がある行列式について、退化条件より、};

&mathjax{=\left| \begin{array}{ccccccc} a_{11} & \dots & \textcolor{blue}{a_{1j}} & \dots & \textcolor{red}{a_{1k}} & \dots & a_{1n} \\ a_{21} & & \textcolor{blue}{a_{2j}} & & \textcolor{red}{a_{2k}} & & a_{2n} \\ \vdots & & \vdots & & \vdots & & \vdots \\a_{n1} & \dots & \textcolor{blue}{a_{nj}} & \dots & \textcolor{red}{a_{nk}} & \dots & a_{nn} \end{array}\right|};


*【P-2】ある列をスカラー倍 [#P2]

これも列の線形多重性の簡易版と考えることができるため、特に計算を要することななく、以下のことが言える。

&mathjax{\left| \begin{array}{ccccc} a_{11} & \dots & \textcolor{red}{c}\textcolor{blue}{a_{1j}} & \dots & a_{1n} \\ a_{21} & & \textcolor{red}{c}\textcolor{blue}{a_{2j}} & & a_{2n} \\ \vdots & & \vdots & & \vdots \\a_{n1} & \dots & \textcolor{red}{c}\textcolor{blue}{a_{nk}} & \dots & a_{nn} \end{array}\right|};&mathjax{=\Large \textcolor{red}{c} \normalsize \left| \begin{array}{ccccc} a_{11} & \dots & \textcolor{blue}{a_{1k}} & \dots & a_{1n} \\ a_{21} & & \textcolor{blue}{a_{2j}} & & a_{2n} \\ \vdots & & \vdots & & \vdots \\a_{n1} & \dots & \textcolor{blue}{a_{nk}} & \dots & a_{nn} \end{array}\right|};

*【P-3】ある列をスカラー倍 [#P3]

&color(Purple){退化条件より、};



&mathjax{\left| \begin{array}{ccccccc} a_{11} & \dots & \overbrace{\textcolor{blue}{a_{1j}}+\textcolor{red}{a_{1k}}}^{\textcolor{blue}{j列目}} & \dots & \overbrace{\textcolor{blue}{a_{1j}}+\textcolor{red}{a_{1k}}}^{\textcolor{red}{k列目}} & \dots & a_{1n} \\ a_{21} & & \textcolor{blue}{a_{2j}}+\textcolor{red}{a_{2k}} & & \textcolor{blue}{a_{2j}}+\textcolor{red}{a_{2k}} & & a_{2n} \\ \vdots & & \vdots & & \vdots & & \vdots \\a_{n1} & \dots & \textcolor{blue}{a_{nj}}+\textcolor{red}{a_{nk}} & \dots & \textcolor{blue}{a_{nj}}+\textcolor{red}{a_{nk}} & \dots & a_{nn} \end{array}\right| \huge = 0 \normalsize};



&color(Purple){線形多重性より};


&mathjax{\left| \begin{array}{ccccccc} a_{11} & \dots & \overbrace{\textcolor{blue}{a_{1j}}}^{\textcolor{blue}{j列目}} & \dots & \overbrace{\textcolor{blue}{a_{1j}}+\textcolor{red}{a_{1k}}}^{\textcolor{red}{k列目}} & \dots & a_{1n} \\ a_{21} & & \textcolor{blue}{a_{2j}} & & \textcolor{blue}{a_{2j}}+\textcolor{red}{a_{2k}} & & a_{2n} \\ \vdots & & \vdots & & \vdots & & \vdots \\a_{n1} & \dots & \textcolor{blue}{a_{nj}} & \dots & \textcolor{blue}{a_{nj}}+\textcolor{red}{a_{nk}} & \dots & a_{nn} \end{array}\right|};&mathjax{+\left| \begin{array}{ccccccc} a_{11} & \dots & \overbrace{\textcolor{red}{a_{1k}}}^{\textcolor{blue}{j列目}} & \dots & \overbrace{\textcolor{blue}{a_{1j}}+\textcolor{red}{a_{1k}}}^{\textcolor{red}{k列目}} & \dots & a_{1n} \\ a_{21} & & \textcolor{red}{a_{2k}} & & \textcolor{blue}{a_{2j}}+\textcolor{red}{a_{2k}} & & a_{2n} \\ \vdots & & \vdots & & \vdots & & \vdots \\a_{n1} & \dots & \textcolor{red}{a_{nk}} & \dots & \textcolor{blue}{a_{nj}}+\textcolor{red}{a_{nk}} & \dots & a_{nn} \end{array}\right| \huge = 0 \normalsize};

&color(Purple){さらに線形多重性より};

&mathjax{\left| \begin{array}{ccccccc} a_{11} & \dots & \overbrace{\textcolor{blue}{a_{1j}}}^{\textcolor{blue}{j列目}} & \dots & \overbrace{\textcolor{blue}{a_{1j}}}^{\textcolor{red}{k列目}} & \dots & a_{1n} \\ a_{21} & & \textcolor{blue}{a_{2j}} & & \textcolor{blue}{a_{2j}} & & a_{2n} \\ \vdots & & \vdots & & \vdots & & \vdots \\a_{n1} & \dots & \textcolor{blue}{a_{nj}} & \dots & \textcolor{blue}{a_{nj}} & \dots & a_{nn} \end{array}\right|};&mathjax{+\left| \begin{array}{ccccccc} a_{11} & \dots & \overbrace{\textcolor{blue}{a_{1j}}}^{\textcolor{blue}{j列目}} & \dots & \overbrace{\textcolor{red}{a_{1k}}}^{\textcolor{red}{k列目}} & \dots & a_{1n} \\ a_{21} & & \textcolor{blue}{a_{2j}} & & \textcolor{red}{a_{2k}} & & a_{2n} \\ \vdots & & \vdots & & \vdots & & \vdots \\a_{n1} & \dots & \textcolor{blue}{a_{nj}} & \dots & \textcolor{red}{a_{nk}} & \dots & a_{nn} \end{array}\right|};&mathjax{+\left| \begin{array}{ccccccc} a_{11} & \dots & \overbrace{\textcolor{red}{a_{1k}}}^{\textcolor{blue}{j列目}} & \dots & \overbrace{\textcolor{blue}{a_{1j}}}^{\textcolor{red}{k列目}} & \dots & a_{1n} \\ a_{21} & & \textcolor{red}{a_{2k}} & & \textcolor{blue}{a_{2j}} & & a_{2n} \\ \vdots & & \vdots & & \vdots & & \vdots \\a_{n1} & \dots & \textcolor{red}{a_{nk}} & \dots & \textcolor{blue}{a_{nj}} & \dots & a_{nn} \end{array}\right|};&mathjax{+\left| \begin{array}{ccccccc} a_{11} & \dots & \overbrace{\textcolor{red}{a_{1k}}}^{\textcolor{blue}{j列目}} & \dots & \overbrace{\textcolor{red}{a_{1k}}}^{\textcolor{red}{k列目}} & \dots & a_{1n} \\ a_{21} & & \textcolor{red}{a_{2k}} & & \textcolor{red}{a_{2k}} & & a_{2n} \\ \vdots & & \vdots & & \vdots & & \vdots \\a_{n1} & \dots & \textcolor{red}{a_{nk}} & \dots & \textcolor{red}{a_{nk}} & \dots & a_{nn} \end{array}\right| \huge = 0 \normalsize};

&color(Purple){退化条件より};



&mathjax{\left| \begin{array}{ccccccc} a_{11} & \dots & \overbrace{\textcolor{blue}{a_{1j}}}^{\textcolor{blue}{j列目}} & \dots & \overbrace{\textcolor{red}{a_{1k}}}^{\textcolor{red}{k列目}} & \dots & a_{1n} \\ a_{21} & & \textcolor{blue}{a_{2j}} & & \textcolor{red}{a_{2k}} & & a_{2n} \\ \vdots & & \vdots & & \vdots & & \vdots \\a_{n1} & \dots & \textcolor{blue}{a_{nj}} & \dots & \textcolor{red}{a_{nk}} & \dots & a_{nn} \end{array}\right|};&mathjax{+\left| \begin{array}{ccccccc} a_{11} & \dots & \overbrace{\textcolor{red}{a_{1k}}}^{\textcolor{blue}{j列目}} & \dots & \overbrace{\textcolor{blue}{a_{1j}}}^{\textcolor{red}{k列目}} & \dots & a_{1n} \\ a_{21} & & \textcolor{red}{a_{2k}} & & \textcolor{blue}{a_{2j}} & & a_{2n} \\ \vdots & & \vdots & & \vdots & & \vdots \\a_{n1} & \dots & \textcolor{red}{a_{nk}} & \dots & \textcolor{blue}{a_{nj}} & \dots & a_{nn} \end{array}\right| \huge = 0 \normalsize};






&color(Purple){よって、};

&mathjax{\left| \begin{array}{ccccccc} a_{11} & \dots & \overbrace{\textcolor{blue}{a_{1j}}}^{\textcolor{blue}{j列目}} & \dots & \overbrace{\textcolor{red}{a_{1k}}}^{\textcolor{red}{k列目}} & \dots & a_{1n} \\ a_{21} & & \textcolor{blue}{a_{2j}} & & \textcolor{red}{a_{2k}} & & a_{2n} \\ \vdots & & \vdots & & \vdots & & \vdots \\a_{n1} & \dots & \textcolor{blue}{a_{nj}} & \dots & \textcolor{red}{a_{nk}} & \dots & a_{nn} \end{array}\right|};&mathjax{\Large = - \normalsize \left| \begin{array}{ccccccc} a_{11} & \dots & \overbrace{\textcolor{red}{a_{1k}}}^{\textcolor{blue}{j列目}} & \dots & \overbrace{\textcolor{blue}{a_{1j}}}^{\textcolor{red}{k列目}} & \dots & a_{1n} \\ a_{21} & & \textcolor{red}{a_{2k}} & & \textcolor{blue}{a_{2j}} & & a_{2n} \\ \vdots & & \vdots & & \vdots & & \vdots \\a_{n1} & \dots & \textcolor{red}{a_{nk}} & \dots & \textcolor{blue}{a_{nj}} & \dots & a_{nn} \end{array}\right|};

つまり、行を入れ替えると、符号が逆になる。