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线性代数之二阶与三阶行业式

2022-07-23 02:46:00 【闲人不梦卿】

二阶与三阶行业式

1.二元线性方程组与二阶行列式

  1.1 如何求解下面的二元线性方程组
{ a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 \left\{ \begin{array}{c} a_{11}x_1+a_{12}x_2=b_1 \\ a_{21}x_1+a_{22}x_2=b_2 \end{array} \right. { a11​x1​+a12​x2​=b1​a21​x1​+a22​x2​=b2​​
  为消除未知数x2，以a22与a12分别乘上列两端方程的两端，然后两个方程相减，得到
( a 11 a 12 − a 12 a 21 ) x 1 = b 1 a 22 − a 12 b 2 \begin{array}{c} (a_{11}a_{12}-a_{12}a_{21})x1=b_1a_{22} -a_{12}b_2\\ \end{array} (a11​a12​−a12​a21​)x1=b1​a22​−a12​b2​​
  类似地，消去x1，得
( a 11 a 12 − a 12 a 21 ) x 2 = a 11 b 2 − b 1 a 21 \begin{array}{c} (a_{11}a_{12}-a_{12}a_{21})x2=a_{11}b_{2} -b_1a_{21}\\ \end{array} (a11​a12​−a12​a21​)x2=a11​b2​−b1​a21​​
  当a11a12-a12a21≠0，求得方程组的解为：
x 1 = b 1 a 22 − a 12 b 2 a 11 a 22 − a 12 a 21 x_1 = \frac {b_1a_{22}-a_{12}b_2} {a_{11}a_{22}-a_{12}a_{21}} x1​=a11​a22​−a12​a21​b1​a22​−a12​b2​​

x 2 = a 11 b 2 − b 1 a 21 a 11 a 22 − a 12 a 21 x_2 = \frac {a_{11}b_2-b_1a_{21}} {a_{11}a_{22}-a_{12}a_{21}} x2​=a11​a22​−a12​a21​a11​b2​−b1​a21​​

  把a11a12-a12a21分母，按照在最上面的二元线性方程组位置，排成二行二列的数表
a 11 a 12 a 21 a 22 \begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{matrix} a11​a21​​a12​a22​​

  表达式a11a12-a12a21称为数表确定的二阶行列式，记作
∣ a 11 a 12 a 21 a 22 ∣ \left| \begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{matrix} \right| ∣∣∣∣​a11​a21​​a12​a22​​∣∣∣∣​

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本文为[闲人不梦卿]所创，转载请带上原文链接，感谢
https://blog.csdn.net/atu1111/article/details/125735093

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