第一章 行列式(知识点部分)

$1.n阶行列式$

代数和 $\sum (-1)^ta_{1p_1}a_{2p_2}...a_{np_n}$称为 $n$阶行列式 $(t为排列p_1p_2...p_n的逆序数)$，记作 $\begin{vmatrix} a_{11}&a_{12}&\dots& a_{1n} \\ a_{21} &a_{22}& \dots&a_{2n} \\ \vdots& \vdots & &\vdots \\ a_{n1} &a_{12}&\dots&a_{nn} \end {vmatrix}$
简记作 $det(a_{ij})$,其中 $a_{ij}$为行列式 $D$的 $(i,j)$元.

$2.主上(下)三角行列式$

$D=\begin{vmatrix} a_{11} \\ a_{21}&a_{22}& &0\\ \vdots&\vdots&\ddots \\ a_{n1}&a_{n2}&\dots&a_{nn} \\ \end{vmatrix}=a_{11}a_{22}{\dots}a_{nn};(对角线元素乘积)$

$3.主对角行列式$

$D=\begin{vmatrix} \lambda_{1} \\ & &\lambda_{2} \\ & & &\ddots \\ & & & &\lambda_{n} \end{vmatrix}=\lambda_{1}\lambda_{2}{\dots}\lambda_{n}.(对角线元素乘积)$

$4.行列式的性质$

$(1)行列式与它的转置行列式相等.$
$D=\left|\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1 n} \\ a_{21} & a_{22} & \cdots & a_{2 n} \\ \vdots & \vdots & & \vdots \\ a_{n 1} & a_{n 2} & \cdots & a_{n n} \end{array}\right|=D^{T}=\left|\begin{array}{cccc} a_{11} & a_{21} & \cdots & a_{n 1} \\ a_{12} & a_{22} & \cdots & a_{n 2} \\ \vdots & \vdots & & \vdots \\ a_{1 n} & a_{2 n} & \cdots & a_{n n} \end{array}\right|$

$(2)对换行列式的两行(列)，行列式变号.$

$(3)行列式中的某一行(列)的所有元素的公因子可以提到行列式记号的外面.$
$\begin{vmatrix} a_{11}&a_{12}&\dots&a_{1n}\\ \vdots&\vdots&&\vdots\\ ka_{i1}&ka_{i2}&\dots&ka_{in}\\ \vdots&\vdots&&\vdots\\ a_{n1}&a_{n2}&\dots&a_{nn}\\ \end{vmatrix}\xlongequal{r_{i}÷k}k\begin{vmatrix} a_{11}&a_{12}&\dots&a_{1n}\\ \vdots&\vdots&&\vdots\\ a_{i1}&a_{i2}&\dots&a_{in}\\ \vdots&\vdots&&\vdots\\ a_{n1}&a_{n2}&\dots&a_{nn}\\ \end{vmatrix}$
$(4)行列式中如果有两行(列)元素成比例,则此行列式等于零.$

$(5)若行列式的某一行(列)的元素都是两数之和，例如第i行的元素都是两数之和:$
$D=\begin{vmatrix} a_{11}&a_{12}&\dots&a_{1n}\\ \vdots&\vdots&&\vdots\\ a_{i1}+a{\prime}_{i1}&a_{i2}+a{\prime}_{i2}&\dots&a_{in}+a{\prime}_{in}\\ \vdots&\vdots&&\vdots\\ a_{n1}&a_{n2}&\dots&a_{nn}\\ \end{vmatrix}$
$则D等于下列两个行列式之和:$
$D=\begin{vmatrix} a_{11}&a_{12}&\dots&a_{1n}\\ \vdots&\vdots&&\vdots\\ a_{i1}&a_{i2}&\dots&a_{in}\\ \vdots&\vdots&&\vdots\\ a_{n1}&a_{n2}&\dots&a_{nn}\\ \end{vmatrix}+\begin{vmatrix} a_{11}&a_{12}&\dots&a_{1n}\\ \vdots&\vdots&&\vdots\\ a{\prime}_{i1}&a{\prime}_{i2}&\dots&a{\prime}_{in}\\ \vdots&\vdots&&\vdots\\ a_{n1}&a_{n2}&\dots&a_{nn}\\ \end{vmatrix}$
$(6)把行列式的某一行(列)的各元素乘同一数然后加到另一行(列)对应的元素上去，行列式不变.$
$\begin{vmatrix} a_{11}&a_{12}&\dots&a_{1n}\\ \vdots&\vdots&&\vdots\\ a_{i1}&a_{i2}&\dots&a_{in}\\ \vdots&\vdots&&\vdots\\ a_{j1}&a_{j2}&\dots&a_{jn}\\ \vdots&\vdots&&\vdots\\ a_{n1}&a_{n2}&\dots&a_{nn}\\ \end{vmatrix}\xlongequal{r_{j}+kr_i}\begin{vmatrix} a_{11}&a_{12}&\dots&a_{1n}\\ \vdots&\vdots&&\vdots\\ a_{i1}&a_{i2}&\dots&a_{in}\\ \vdots&\vdots&&\vdots\\ a_{j1}+ka_{i1}&a_{j2}+ka_{i2}&\dots&a_{jn}+ka_{in}\\ \vdots&\vdots&&\vdots\\ a_{n1}&a_{n2}&\dots&a_{nn}\\ \end{vmatrix} (i\not = j)$

$5.副上(下)三角行列式$

$6.副对角行列式$

$7.D=D_1D_2$

$D=\left|\begin{array}{ccccc} a_{11} & \cdots & a_{1 k} & & \\ \vdots & &\vdots & & 0 & \\ a_{k 1} & \cdots & a_{k k} & & & \\ c_{11} & \cdots & c_{1 k} & b_{11} & \cdots & b_{1 n} \\ \vdots & & \vdots & \vdots & & \vdots \\ c_{n 1} & \cdots & c_{n k} & b_{n 1} & \cdots & b_{n n} \end{array}\right|$
$D_1=\begin{vmatrix} a_{11}&\dots&a_{1k}\\ \vdots&&\vdots\\ a_{k1}&\dots&a_{kk}\\ \end{vmatrix},D_2=\begin{vmatrix} b_{11}&\dots&b_{1n}\\ \vdots&&\vdots\\ b_{n1}&\dots&b_{nn}\\ \end{vmatrix}$

$8.X型行列式$

$9.范德蒙德行列式$

$D_{n}=\left|\begin{array}{cccc} 1 & 1 & \cdots & 1 \\ x_{1} & x_{2} & \cdots & x_{n} \\ x_{1}^{2} & x_{2}^{2} & \cdots & x_{n}^{2} \\ \vdots & \vdots & & \vdots \\ x_{1}^{n-1} & x_{2}^{n-1} & \cdots & x_{n}^{n-1} \end{array}\right|=\prod_{n \geq 1 \geq j \geq 1}\left(x_{i-} x_{j}\right)$

$10.行列式按行(列)展开法则及零值定理$

$\sum_{k=1}^{n} a_{i k} A_{j k}=\left\{\begin{array}{c} D, 当i=j,\\ 0, 当i\not = j,\\ \end{array}\right.$