行列式

• 2元連立1次方程式の解
  $$\begin{eqnarray*} a_{11}x_{1}+a_{12}x_{2}&=& b_{1}\\ a_{21}x_{1}+a_{22}x_{2}&=& b_{2} \end{eqnarray*}$$
  
  
  $$\begin{eqnarray*} x_{1}&=& \frac{1}{a_{11}a_{22}-a_{12}a_{21}}(a_{22}b_{1}-a_{1... ...=& \frac{1}{a_{11}a_{22}-a_{12}a_{21}}(-a_{21}b_{1}+a_{11}b_{2}) \end{eqnarray*}$$
  
  

• 3元連立1次方程式の解
  $$\begin{eqnarray*} a_{11}x_{1}+a_{12}x_{2}+a_{13}x_{3}&=& b_{1}\\ a_{21}x_{1}+... ...}x_{3}&=& b_{2}\\ a_{31}x_{1}+a_{32}x_{2}+a_{33}x_{3}&=& b_{3} \end{eqnarray*}$$
  
  
  $$\begin{eqnarray*} A&=& a_{11}a_{22}a_{33}-a_{11}a_{23}a_{32} \\ && -a_{12}a_{... ..._{12}a_{23}a_{31} \\ && -a_{13}a_{22}a_{31}+a_{13}a_{21}a_{32} \end{eqnarray*}$$
  
  
  $$\begin{eqnarray*} x_{1}&=& \frac{1}{A} \{(a_{22}a_{33}-a_{23}a_{31})b_1 \\ &&... ...eft\{\cdots\right\}\\ x_{3}&=&\frac{1}{A}\left\{\cdots\right\} \end{eqnarray*}$$
  
  

連立1次方程式の解の分母に現れる式を行列式と呼ぶ:
2次の行列式

$$\begin{eqnarray*} \left\vert \begin{array}{cc} a_{11}& a_{12}\\ a_{21}& a_{22} \end{array}\right\vert &=& a_{11}a{22}-a_{12}a_{21} \end{eqnarray*}$$

3次の行列式

$$\begin{eqnarray*} \left\vert\begin{array}{ccc} a_{11}& a_{12}& a_{13}\\ a_{2... ...a_{12}a_{23}a_{31} \\ && -a_{13}a_{22}a_{31}+a_{13}a_{21}a_{32} \end{eqnarray*}$$