二阶三阶行列式

“行数等于列数”，边界有两条竖线的算式，结果是个“数”

D = a_{11} a_{21} ⋮ a_{n 1} a_{12} a_{22} ⋮ a_{n 2} \dots \dots ⋱ \dots a_{1 n} a_{2 n} ⋮ a_{nn}

共有n行n列， $n^{2}$ 个数（元素）。对一个行列式求导结果为0

{3 x - 2 y = 12 2 x + y = 1 抽像化 {a_{11} x_{1} + a_{12} x_{2} = b_{1} a_{21} x_{1} + a_{22} x_{2} = b_{2} ① ②

①乘以 $a_{22}$
②乘以 $a_{12}$

{a_{11} a_{22} x_{1} + a_{12} a_{22} x_{2} = b_{1} a_{22} a_{21} a_{12} x_{1} + a_{22} a_{12} x_{2} = b_{2} a_{12}

$(a_{11} a_{22} - a_{21} a_{12}) x_{1} = b_{1} a_{22} - b_{2} a_{12}$
$当 a_{11} a_{22} - a_{21} a_{12} \neq = 0$

x_{1} = \frac{b _{1} a _{22} - b _{2} a _{12}}{a _{11} a _{22} - a _{21} a _{12}}, x_{2} = \frac{a _{11} b _{2} - a _{22} b _{1}}{a _{11} a _{22} - a _{12} a _{21}}

二阶行列式

行列式: determinant

D = a_{11} a_{21} a_{12} a_{22} = a_{11} a_{22} - a_{12} a_{21}

当上面的方程组的未知数前的系数构成的行列式结果不为0，决定这个方程组具有唯一解
$D = a_{11} a_{21} a_{12} a_{22} \neq = 0 ⟹ 方程组具有唯一解$

D = a_{11} a_{21} a_{12} a_{22} = 0 ⟹ 方程组具有无穷解，或无解

比如说：

{x_{1} + x_{2} = 1 x_{1} + x_{2} = - 1 {x_{1} + x_{2} = 1 x_{1} + x_{2} = 1

化简

x_{1} = \frac{b _{1} a _{22} - b _{2} a _{12}}{a _{11} a _{22} - a _{21} a _{12}}, x_{2} = \frac{a _{11} b _{2} - a _{22} b _{1}}{a _{11} a _{22} - a _{12} a _{21}}

x_{1} = \frac{b _{1} a _{22} - b _{2} a _{12}}{D}, x_{2} = \frac{a _{11} b _{2} - a _{22} b _{1}}{D}

把常数项换到第一列

b_{1} a_{22} - b_{2} a_{12} = b_{1} a_{12} b_{2} a_{22} = D_{1}

a_{11} b_{1} a_{21} b_{2} = a_{11} b_{2} - a_{22} b_{1} = D_{2}

x_{1}=\frac{D_1}D,x_{2}=\frac{D_2}D