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 \(\left|\begin{array}{c} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right|\) 的式子,它表示一个数值,其展开式为
\[\left|\begin{array}{c} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right| =a_{11}a_{22}-a_{12}a_{21} \]
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 \(\left|\begin{array}{c} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right|\) 的式子,它表示
一个数值,其展开式为
\[\left|\begin{array}{c} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right| =a_{11}\left|\begin{array}{c} a_{22} & a_{23} \\ a_{32} & a_{33} \end{array}\right|-a_{12} \left|\begin{array}{c} a_{21} & a_{23} \\ a_{31} & a_{33} \end{array}\right|+a_{13} \left|\begin{array}{c} a_{21} & a_{22} \\ a_{31} & a_{32} \end{array}\right| \]
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 \(\left|\begin{array}{c} a_{11} \end{array}\right|\) 的式子,它表示一个数值,其展开式为
\[\left|\begin{array}{c} a_{11} \end{array}\right| =a_{11} \]
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:
\[\left|\begin{array}{c} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right| =a_{11}\left|\begin{array}{c} a_{22} \end{array}\right|-a_{12}\left|\begin{array}{c} a_{21} \end{array}\right| \]
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 \(n\) 行 \(n\) 列形如 \(\left|\begin{array}{c} a_{11} & \cdots & a_{1n} \\ \cdots & \ddots & \cdots \\ a_{n1} & \cdots & a_{nn} \end{array}\right|\) 的式子,它表示一个数值,其展开式为
\[\left|\begin{array}{c} a_{11} & \cdots & a_{1n} \\ \cdots & \ddots & \cdots \\ a_{n1} & \cdots & a_{nn} \end{array}\right| =\sum_{i=1}^{n}(-1)^{i+1}a_{1i}\left|\begin{array}{c} a_{21} & \cdots & a_{2\ i-1} & a_{2\ i+1} & \cdots & a_{2n} \\ \cdots & \ddots & \ddots & \ddots & \ddots & \cdots \\ \cdots & \ddots & \ddots & \ddots & \ddots & \cdots \\ \cdots & \ddots & \ddots & \ddots & \ddots & \cdots \\ \cdots & \ddots & \ddots & \ddots & \ddots & \cdots \\ a_{n1} & \cdots & a_{n\ i-1} & a_{n\ i+1} & \cdots & a_{nn} \end{array}\right| \]
(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|\begin{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}\right|$$,称为元素 $a_{1i}$ 的**余子式**。令 \]
A_{1i}=(-1)^{i+1}M_{1i}$$,称为元素 \(a_{1j}\) 的代数余子式。
行列式在解线性方程的运用:Cramer法则
目标:求解关于 \(x_1\),\(x_2\),\(\cdots\),\(x_n\) 的 \(n\) 元线性方程组
\[\begin{cases} a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n = b_1 \\ a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n = b_2\\ \cdots \\ a_{n1}x_1 + a_{n2}x_2 + \cdots + a_{nn}x_n = b_n \\ \end{cases} \]
Cramer法则求解:
令
\[D=\left|\begin{array}{c} a_{11} & \cdots & a_{1n} \\ \cdots & \ddots & \cdots \\ a_{n1} & \cdots & a_{nn} \end{array}\right| \]
,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
\[D_1=\left|\begin{array}{c} b_1 & a_{12} & \cdots & a_{1n} \\ b_2 & a_{22} & \cdots & a_{2n} \\ \cdots & \vdots & \ddots & \cdots \\ b_n & a_{n2} & \cdots & a_{nn} \end{array}\right|, D_2=\left|\begin{array}{c} a_{11} & b_1 & \cdots & a_{1n} \\ a_{21} & b_2 & \cdots & a_{2n} \\ \cdots & \vdots & \ddots & \cdots \\ a_{n1} & b_n & \cdots & a_{nn} \end{array}\right|, \cdots, D_n=\left|\begin{array}{c} a_{11} & a_{12} & \cdots & b_1 \\ a_{21} & a_{22} & \cdots & b_2 \\ \cdots & \vdots & \ddots & \cdots \\ a_{n1} & a_{n2} & \cdots & b_n \end{array}\right| \]
若线性方程组的系数行列式 \(D\not=0\),则该方程组有唯一解:
\[x_i=D/D_i\qquad (i=1,2,\cdots,n) \]
Cramer法则的应用
例题 求解二元线性方程组
\[\begin{cases} 5x_1+x_2 = 4 \\ 2x_1-3x_2 = 5 \end{cases} \]
解 这个线性方程组的系数行列式为
\[D=\left|\begin{array}{c} 5 & 1 \\ 2 & -3 \end{array}\right|=-17 \]
由于 \(D=17\not=0\),该线性方程组有唯一解,
\[D_1=\left|\begin{array}{c} 4 & 1 \\ 5 & -3 \end{array}\right|=-17, D_2=\left|\begin{array}{c} 5 & 4 \\ 2 & 5 \end{array}\right|=17 \]
即
\[\begin{cases} x_1=D/D_1=1 \\ x_2=D/D_2=-1 \end{cases} \]
Cramer法则与齐次性
若线性方程组的常数项全为零,即
\[\begin{cases} a_{11}x_1 + a_{12}x_2 + \cdots + a_{1n}x_n = 0 \\ a_{21}x_1 + a_{22}x_2 + \cdots + a_{2n}x_n = 0 \\ \cdots \\ a_{n1}x_1 + a_{n2}x_2 + \cdots + a_{nn}x_n = 0 \\ \end{cases} \]
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 \(x_i = 0\qquad (i=1,\cdots,n)\),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
- 应用Cramer法则求解 \(n\) 元线性方程组时,必须有 \(n\) 条方程。
- 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 \(n+1\) 个 \(n\) 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
\[D=\left|\begin{array}{c} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \cdots & \vdots & \ddots & \cdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{array}\right| \]
其转置为
\[D^T=\left|\begin{array}{c} a_{11} & a_{21} & \cdots & a_{n1} \\ a_{12} & a_{22} & \cdots & a_{n2} \\ \cdots & \vdots & \ddots & \cdots \\ a_{1n} & a_{2n} & \cdots & a_{nn} \end{array}\right| \]
性质1 \(D\) = \(D^T\)
推论 行列式可按任一行(列)展开,即
\[D=\left|\begin{array}{c} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \cdots & \vdots & \ddots & \cdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{array}\right| =\sum_{j=1}^na_{ij}A_{ij} \]
(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
\[D=\left|\begin{array}{c} a_{11} & a_{12} & \cdots & a_{1n} \\ \cdots & \vdots & \ddots & \cdots \\ ka_{i1} & ka_{i2} & \cdots & ka_{in} \\ \cdots & \vdots & \ddots & \cdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{array}\right| =k \left|\begin{array}{c} a_{11} & a_{12} & \cdots & a_{1n} \\ \cdots & \vdots & \ddots & \cdots \\ a_{i1} & a_{i2} & \cdots & a_{in} \\ \cdots & \vdots & \ddots & \cdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{array}\right| \]
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
\[D=\left|\begin{array}{c} a_{11} & a_{12} & \cdots & a_{1n} \\ \cdots & \vdots & \ddots & \cdots \\ a_{i1} & a_{i2} & \cdots & a_{in} \\ \cdots & \vdots & \ddots & \cdots \\ a_{j1} & a_{j2} & \cdots & a_{jn} \\ \cdots & \vdots & \ddots & \cdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{array}\right| =-\left|\begin{array}{c} a_{11} & a_{12} & \cdots & a_{1n} \\ \cdots & \vdots & \ddots & \cdots \\ a_{j1} & a_{j2} & \cdots & a_{jn} \\ \cdots & \vdots & \ddots & \cdots \\ a_{i1} & a_{i2} & \cdots & a_{in} \\ \cdots & \vdots & \ddots & \cdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{array}\right| \]
性质5 行列式可以拆行(列)相加,即
\[D=\left|\begin{array}{c} a_{11} & a_{12} & \cdots & a_{1n} \\ \cdots & \vdots & \ddots & \cdots \\ a_{i1}+a'_{i1} & a_{i2}+a'_{i2} & \cdots & a_{in}+a'_{in} \\ \cdots & \vdots & \ddots & \cdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{array}\right| =\left|\begin{array}{c} a_{11} & a_{12} & \cdots & a_{1n} \\ \cdots & \vdots & \ddots & \cdots \\ a_{i1} & a_{i2} & \cdots & a_{in} \\ \cdots & \vdots & \ddots & \cdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{array}\right| +\left|\begin{array}{c} a_{11} & a_{12} & \cdots & a_{1n} \\ \cdots & \vdots & \ddots & \cdots \\ a'_{i1} & a'_{i2} & \cdots & a'_{in} \\ \cdots & \vdots & \ddots & \cdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{array}\right| \]
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
\[D=\left|\begin{array}{c} a_{11} & a_{12} & \cdots & a_{1n} \\ \cdots & \vdots & \ddots & \cdots \\ a_{i1} & a_{i2} & \cdots & a_{in} \\ \cdots & \vdots & \ddots & \cdots \\ a_{j1} & a_{j2} & \cdots & a_{jn} \\ \cdots & \vdots & \ddots & \cdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{array}\right| =\left|\begin{array}{c} a_{11} & a_{12} & \cdots & a_{1n} \\ \cdots & \vdots & \ddots & \cdots \\ a_{j1} & a_{j2} & \cdots & a_{jn} \\ \cdots & \vdots & \ddots & \cdots \\ a_{i1}+a_{j1} & a_{i2}+a_{j2} & \cdots & a_{in}+a_{jn} \\ \cdots & \vdots & \ddots & \cdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{array}\right| \]