行列式的向量形式

行列式公式

\[|A| = \begin{vmatrix} a_{11} & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22} & \cdots & a_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ a_{n1} & a_{n2} & \cdots & a_{nn}\\ \end{vmatrix}\]

行向量表示

令\(\alpha_i = (a_{i1} , a_{i2} , \cdots , a_{in}), i\in(1,2,\cdots,n)\), 則行列式可以表示為

\[|A| = \begin{vmatrix} a_{11} & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22} & \cdots & a_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ a_{n1} & a_{n2} & \cdots & a_{nn}\\ \end{vmatrix} = \begin{vmatrix} \alpha_1 \\ \alpha_2 \\ \vdots \\ \alpha_n \end{vmatrix}\]

性質2(0向量)

\[|A|= \begin{vmatrix} \alpha_1 \\ \vdots \\ 0 \\ \vdots \\ \alpha_n \end{vmatrix} =0\]

性質3(某一向量的倍數)

\[|B|= \begin{vmatrix} \alpha_1 \\ \vdots \\ c\alpha_i \\ \vdots \\ \alpha_n \end{vmatrix} =c\begin{vmatrix} \alpha_1 \\ \vdots \\ \alpha_i \\ \vdots \\ \alpha_n \end{vmatrix} = c|A|\]

性質4(兩行互換)

\[|B|= \begin{vmatrix} \alpha_1 \\ \vdots \\ \alpha_j \\ \vdots \\ \alpha_i \\ \vdots \\ \alpha_n \end{vmatrix} = -\begin{vmatrix} \alpha_1 \\ \vdots \\ \alpha_i \\ \vdots \\ \alpha_j \\ \vdots \\ \alpha_n \end{vmatrix} = -|A|\]

性質5(向量加法c=a+b)

\[|C|= \begin{vmatrix} \alpha_1 \\ \vdots \\ a+b \\ \vdots \\ \alpha_n \end{vmatrix} =\begin{vmatrix} \alpha_1 \\ \vdots \\ a \\ \vdots \\ \alpha_n \end{vmatrix} +\begin{vmatrix} \alpha_1 \\ \vdots \\ b \\ \vdots \\ \alpha_n \end{vmatrix} =|A|+|B|\]

性質6(兩行成比例)

\[|B|= \begin{vmatrix} \alpha_1 \\ \vdots \\ a \\ \vdots \\ ca \\ \vdots \\ \alpha_n \end{vmatrix} = c\begin{vmatrix} \alpha_1 \\ \vdots \\ a \\ \vdots \\ a \\ \vdots \\ \alpha_n \end{vmatrix} = 0\]

性質7(倍加)

\[|B|= \begin{vmatrix} \alpha_1 \\ \vdots \\ a \\ \vdots \\ ca+b \\ \vdots \\ \alpha_n \end{vmatrix} = \begin{vmatrix} \alpha_1 \\ \vdots \\ a \\ \vdots \\ ca \\ \vdots \\ \alpha_n \end{vmatrix} + \begin{vmatrix} \alpha_1 \\ \vdots \\ a \\ \vdots \\ b \\ \vdots \\ \alpha_n \end{vmatrix} = |A|\]

列向量表示

令\(\beta_i = \begin{pmatrix}a_{1i} \\ a_{2i} \\ \vdots \\ a_{ni}\end{pmatrix}, i\in(1,2,\cdots,n)\), 則行列式可以表示為

\[|A| = \begin{vmatrix} a_{11} & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22} & \cdots & a_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ a_{n1} & a_{n2} & \cdots & a_{nn}\\ \end{vmatrix} = \begin{vmatrix} \beta_1 & \beta_2 & \cdots & \beta_n \end{vmatrix}\]

性質2(0向量)

\[|A|= \begin{vmatrix} \beta_1 & \cdots & 0 & \cdots & \beta_n \end{vmatrix} =0\]

性質3(某一向量的倍數)

\[|B|= \begin{vmatrix} \beta_1 & \cdots & c\beta_i & \cdots & \beta_n \end{vmatrix} \]

\[=c\begin{vmatrix} \beta_1 & \cdots & \beta_i & \cdots & \beta_n \end{vmatrix} \]

\[= c|A| \]

性質4(兩行互換)

\[|B|= \begin{vmatrix} \beta_1 & \cdots & \beta_j & \cdots & \beta_i & \cdots & \beta_n \end{vmatrix}\]

\[= -\begin{vmatrix} \beta_1 & \cdots & \beta_i & \cdots & \beta_j & \cdots & \beta_n \end{vmatrix} \]

\[= -|A| \]

性質5(向量加法c=a+b)

\[|C|= \begin{vmatrix} \beta_1 & \cdots & a+b & \cdots & \beta_n \end{vmatrix} \]

\[=\begin{vmatrix} \beta_1 & \cdots & a & \cdots & \beta_n \end{vmatrix} +\begin{vmatrix} \beta_1 & \cdots & b & \cdots & \beta_n \end{vmatrix} \]

\[=|A|+|B| \]

性質6(兩行成比例)

\[|B|= \begin{vmatrix} \beta_1 & \cdots & a & \cdots & ca & \cdots & \beta_n \end{vmatrix} \]

\[= c\begin{vmatrix} \beta_1 & \cdots & a & \cdots & a & \cdots & \beta_n \end{vmatrix} \]

\[= 0 \]

性質7(倍加)

\[|B|= \begin{vmatrix} \beta_1 & \cdots & a & \cdots & ca+b & \cdots & \beta_n \end{vmatrix} \]

\[= \begin{vmatrix} \beta_1 & \cdots & a & \cdots & ca & \cdots & \beta_n \end{vmatrix} + \begin{vmatrix} \beta_1 & \cdots & a & \cdots & b & \cdots & \beta_n \end{vmatrix} \]

\[= |A| \]