线性代数->行列式->二阶与三阶行列式
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一、二元线性方程组与二阶行列式
用消元法解二元线性方程组
$$ f(x)=\left\{ \begin{aligned} a_{11}x_1+a_{12}x_2 &=b_1 \\ a_{21}x_1+a_{22}x_2 &=b_2 \end{aligned} \right.$$
当$a_{11}a_{22}-a_{12}a_{21}\not=0$时，求得方程组的解为
$x_1=\cfrac{b_1 a_{22}-a_{12}b_2}{a_{11}a_{22}-a_{12}a_{21}},\quad x_2=\cfrac{a_{11}b_2-b_1a_{21}}{a_{11}a_{22}-a_{12}a_{21}}$
$\begin{equation} \left | \begin{array}{cc} a_{11}& a_{12} \\ a_{21}& a_{22} \end{array} \right | \end{equation}$
表达式$a_{11}a_{22}-a_{12}a_{21}$称为二阶行列式，记为上述形式。
若记
$D=\begin{equation} \left | \begin{array}{cc} a_{11}& a_{12} \\ a_{21}& a_{22} \end{array} \right | \end{equation}$
$D_1=\begin{equation} \left | \begin{array}{cc} b_{1}& a_{12} \\ b_{2}& a_{22} \end{array} \right | \end{equation}$
$D_2=\begin{equation} \left | \begin{array}{cc} a_{11}& b_{1} \\ a_{21}& b_{2} \end{array} \right | \end{equation}$
则$x_1,x_2$可表述为
$x_1=\cfrac{D_1}{D},\quad x_2=\cfrac{D_2}{D}$