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  • 线性代数1 行列式

    二阶行列式

    所谓二阶行列式,是由四个数,如 (a_{11})(a_{12})(a_{21})(a_{22}) 排列成含有两行两列形如 (left|egin{array}{c} a_{11} & a_{12} \ a_{21} & a_{22} end{array} ight|) 的式子,它表示一个数值,其展开式为

    [left|egin{array}{c} a_{11} & a_{12} \ a_{21} & a_{22} end{array} ight| =a_{11}a_{22}-a_{12}a_{21} ]

    三阶行列式

    所谓三阶行列式,是由九个数,如 (a_{11})(a_{12})(a_{13})(a_{21})(a_{22})(a_{23})(a_{31})(a_{32})(a_{33}) 排列成含有三行三列形如 (left|egin{array}{c} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} end{array} ight|) 的式子,它表示

    一个数值,其展开式为

    [left|egin{array}{c} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} end{array} ight| =a_{11}left|egin{array}{c} a_{22} & a_{23} \ a_{32} & a_{33} end{array} ight|-a_{12} left|egin{array}{c} a_{21} & a_{23} \ a_{31} & a_{33} end{array} ight|+a_{13} left|egin{array}{c} a_{21} & a_{22} \ a_{31} & a_{32} end{array} ight| ]

    n阶行列式

    我们观察二、三阶行列式的定义,顺便定义一下一阶行列式:

    (几乎全是复制)

    所谓一阶行列式,是由一个数,如 (a_{11}) 排列成含有一行一列形如 (left|egin{array}{c} a_{11} end{array} ight|) 的式子,它表示一个数值,其展开式为

    [left|egin{array}{c} a_{11} end{array} ight| =a_{11} ]

    有了一阶行列式的定义,我们考虑像三阶行列式一样递归的定义二阶行列式:

    [left|egin{array}{c} a_{11} & a_{12} \ a_{21} & a_{22} end{array} ight| =a_{11}left|egin{array}{c} a_{22} end{array} ight|-a_{12}left|egin{array}{c} a_{21} end{array} ight| ]

    至此,(n) 阶行列式的定义几乎呼之欲出了:
    所谓 (n) 阶行列式,是由 (n^2) 个数,如 (a_{11})(a_{12})(cdots)(a_{nn}) 排列成含有 (n)(n) 列形如 (left|egin{array}{c} a_{11} & cdots & a_{1n} \ cdots & ddots & cdots \ a_{n1} & cdots & a_{nn} end{array} ight|) 的式子,它表示一个数值,其展开式为

    [left|egin{array}{c} a_{11} & cdots & a_{1n} \ cdots & ddots & cdots \ a_{n1} & cdots & a_{nn} end{array} ight| =sum_{i=1}^{n}(-1)^{i+1}a_{1i}left|egin{array}{c} a_{21} & cdots & a_{2 i-1} & a_{2 i+1} & cdots & a_{2n} \ cdots & ddots & ddots & ddots & ddots & cdots \ cdots & ddots & ddots & ddots & ddots & cdots \ cdots & ddots & ddots & ddots & ddots & cdots \ cdots & ddots & ddots & ddots & ddots & cdots \ a_{n1} & cdots & a_{n i-1} & a_{n i+1} & cdots & a_{nn} end{array} ight| ]

    (其实就是对于第一行的每个元素,用它乘除了它同行同列的剩下来数构成的子行列式。)

    上式中令

    [M_{1i}= left|egin{array}{c} a_{21} & cdots & a_{2 i-1} & a_{2 i+1} & cdots & a_{2n} \ cdots & ddots & ddots & ddots & ddots & cdots \ cdots & ddots & ddots & ddots & ddots & cdots \ cdots & ddots & ddots & ddots & ddots & cdots \ cdots & ddots & ddots & ddots & ddots & cdots \ a_{n1} & cdots & a_{n i-1} & a_{n i+1} & cdots & a_{nn} end{array} ight|$$,称为元素 $a_{1i}$ 的**余子式**。令 ]

    A_{1i}=(-1)^{i+1}M_{1i}$$,称为元素 (a_{1j})代数余子式

    行列式在解线性方程的运用:Cramer法则

    目标:求解关于 (x_1)(x_2)(cdots)(x_n)(n) 元线性方程组

    [egin{cases} a_{11}x_1 + a_{12}x_2 + cdots + a_{1n}x_n = b_1 \ a_{21}x_1 + a_{22}x_2 + cdots + a_{2n}x_n = b_2\ cdots \ a_{n1}x_1 + a_{n2}x_2 + cdots + a_{nn}x_n = b_n \ end{cases} ]

    Cramer法则求解

    [D=left|egin{array}{c} a_{11} & cdots & a_{1n} \ cdots & ddots & cdots \ a_{n1} & cdots & a_{nn} end{array} ight| ]

    ,称之为该方程组的系数行列式

    同时,把行列式 (D) 的第 (i) 列替换为方程组的常数列项((b_1)(b_2)(cdots)(b_n)),得到新的行列式记为 (D_i),即

    [D_1=left|egin{array}{c} b_1 & a_{12} & cdots & a_{1n} \ b_2 & a_{22} & cdots & a_{2n} \ cdots & vdots & ddots & cdots \ b_n & a_{n2} & cdots & a_{nn} end{array} ight|, D_2=left|egin{array}{c} a_{11} & b_1 & cdots & a_{1n} \ a_{21} & b_2 & cdots & a_{2n} \ cdots & vdots & ddots & cdots \ a_{n1} & b_n & cdots & a_{nn} end{array} ight|, cdots, D_n=left|egin{array}{c} a_{11} & a_{12} & cdots & b_1 \ a_{21} & a_{22} & cdots & b_2 \ cdots & vdots & ddots & cdots \ a_{n1} & a_{n2} & cdots & b_n end{array} ight| ]

    若线性方程组的系数行列式 (D ot=0),则该方程组有唯一解

    [x_i=D/D_iqquad (i=1,2,cdots,n) ]

    Cramer法则的应用

    例题 求解二元线性方程组

    [egin{cases} 5x_1+x_2 = 4 \ 2x_1-3x_2 = 5 end{cases} ]

    这个线性方程组的系数行列式为

    [D=left|egin{array}{c} 5 & 1 \ 2 & -3 end{array} ight|=-17 ]

    由于 (D=17 ot=0),该线性方程组有唯一解,

    [D_1=left|egin{array}{c} 4 & 1 \ 5 & -3 end{array} ight|=-17, D_2=left|egin{array}{c} 5 & 4 \ 2 & 5 end{array} ight|=17 ]

    [egin{cases} x_1=D/D_1=1 \ x_2=D/D_2=-1 end{cases} ]

    Cramer法则与齐次性

    若线性方程组的常数项全为零,即

    [egin{cases} a_{11}x_1 + a_{12}x_2 + cdots + a_{1n}x_n = 0 \ a_{21}x_1 + a_{22}x_2 + cdots + a_{2n}x_n = 0 \ cdots \ a_{n1}x_1 + a_{n2}x_2 + cdots + a_{nn}x_n = 0 \ end{cases} ]

    则称该线性方程组为齐次线性方程组。反之,如果常数项不全为零,则称之为非齐次线性方程组

    齐次线性方程组永远有解,这组解为 (x_i = 0qquad (i=1,cdots,n)),这组解被称为零解
    由Cramer法则容易知道,当线性方程的系数行列式不等于 (0) 时,方程只有零解。

    Cramer法则的局限性

    1. 应用Cramer法则求解 (n) 元线性方程组时,必须有 (n) 条方程。
    2. 应用Cramer法则求解 (n) 元线性方程组时,因涉及到行列式的计算问题,即需要计算 (n+1)(n) 阶行列式的值,这样,随着 (n) 的增大,求解的计算量是相当大的。

    行列式的性质

    行列式转置

    对于行列式

    [D=left|egin{array}{c} a_{11} & a_{12} & cdots & a_{1n} \ a_{21} & a_{22} & cdots & a_{2n} \ cdots & vdots & ddots & cdots \ a_{n1} & a_{n2} & cdots & a_{nn} end{array} ight| ]

    其转置为

    [D^T=left|egin{array}{c} a_{11} & a_{21} & cdots & a_{n1} \ a_{12} & a_{22} & cdots & a_{n2} \ cdots & vdots & ddots & cdots \ a_{1n} & a_{2n} & cdots & a_{nn} end{array} ight| ]

    性质1 (D) = (D^T)
    推论 行列式可按任一行(列)展开,即

    [D=left|egin{array}{c} a_{11} & a_{12} & cdots & a_{1n} \ a_{21} & a_{22} & cdots & a_{2n} \ cdots & vdots & ddots & cdots \ a_{n1} & a_{n2} & cdots & a_{nn} end{array} ight| =sum_{j=1}^na_{ij}A_{ij} ]

    (其中 (A_{ij})即为上文所提到的代数余子式。)

    性质2 行列式可以按行(列)提取公因子,即

    [D=left|egin{array}{c} a_{11} & a_{12} & cdots & a_{1n} \ cdots & vdots & ddots & cdots \ ka_{i1} & ka_{i2} & cdots & ka_{in} \ cdots & vdots & ddots & cdots \ a_{n1} & a_{n2} & cdots & a_{nn} end{array} ight| =k left|egin{array}{c} a_{11} & a_{12} & cdots & a_{1n} \ cdots & vdots & ddots & cdots \ a_{i1} & a_{i2} & cdots & a_{in} \ cdots & vdots & ddots & cdots \ a_{n1} & a_{n2} & cdots & a_{nn} end{array} ight| ]

    性质3 行列式中某一行(列)元素全为零时,值为零。
    性质4 行列式两行(列)互换值反号,即

    [D=left|egin{array}{c} a_{11} & a_{12} & cdots & a_{1n} \ cdots & vdots & ddots & cdots \ a_{i1} & a_{i2} & cdots & a_{in} \ cdots & vdots & ddots & cdots \ a_{j1} & a_{j2} & cdots & a_{jn} \ cdots & vdots & ddots & cdots \ a_{n1} & a_{n2} & cdots & a_{nn} end{array} ight| =-left|egin{array}{c} a_{11} & a_{12} & cdots & a_{1n} \ cdots & vdots & ddots & cdots \ a_{j1} & a_{j2} & cdots & a_{jn} \ cdots & vdots & ddots & cdots \ a_{i1} & a_{i2} & cdots & a_{in} \ cdots & vdots & ddots & cdots \ a_{n1} & a_{n2} & cdots & a_{nn} end{array} ight| ]

    性质5 行列式可以拆行(列)相加,即

    [D=left|egin{array}{c} a_{11} & a_{12} & cdots & a_{1n} \ cdots & vdots & ddots & cdots \ a_{i1}+a'_{i1} & a_{i2}+a'_{i2} & cdots & a_{in}+a'_{in} \ cdots & vdots & ddots & cdots \ a_{n1} & a_{n2} & cdots & a_{nn} end{array} ight| =left|egin{array}{c} a_{11} & a_{12} & cdots & a_{1n} \ cdots & vdots & ddots & cdots \ a_{i1} & a_{i2} & cdots & a_{in} \ cdots & vdots & ddots & cdots \ a_{n1} & a_{n2} & cdots & a_{nn} end{array} ight| +left|egin{array}{c} a_{11} & a_{12} & cdots & a_{1n} \ cdots & vdots & ddots & cdots \ a'_{i1} & a'_{i2} & cdots & a'_{in} \ cdots & vdots & ddots & cdots \ a_{n1} & a_{n2} & cdots & a_{nn} end{array} ight| ]

    性质6 行列式两行(列)成比例值为零。
    推论 行列式两行(列)相同值为零。
    性质7 行列式某行(列)的倍数加到另一行(列)值不变,即

    [D=left|egin{array}{c} a_{11} & a_{12} & cdots & a_{1n} \ cdots & vdots & ddots & cdots \ a_{i1} & a_{i2} & cdots & a_{in} \ cdots & vdots & ddots & cdots \ a_{j1} & a_{j2} & cdots & a_{jn} \ cdots & vdots & ddots & cdots \ a_{n1} & a_{n2} & cdots & a_{nn} end{array} ight| =left|egin{array}{c} a_{11} & a_{12} & cdots & a_{1n} \ cdots & vdots & ddots & cdots \ a_{j1} & a_{j2} & cdots & a_{jn} \ cdots & vdots & ddots & cdots \ a_{i1}+a_{j1} & a_{i2}+a_{j2} & cdots & a_{in}+a_{jn} \ cdots & vdots & ddots & cdots \ a_{n1} & a_{n2} & cdots & a_{nn} end{array} ight| ]

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  • 原文地址:https://www.cnblogs.com/szdytom/p/linear-det.html
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