Section02_易算行列式

1. 对角线，下三角，上三角 $$\left\vert \begin{matrix} a_{11} & & & \huge0 \\ & a_{22} & & \\ & & \ddots & \\ \huge0 & & & a_{nn} \\ \end{matrix} \right\vert= \left\vert \begin{matrix} a_{11} & & & \huge0 \\ & a_{22} & & \\ & & \ddots & \\ \huge * & & & a_{nn} \\ \end{matrix} \right\vert = \left\vert \begin{matrix} a_{11} & & & \huge * \\ & a_{22} & & \\ & & \ddots & \\ \huge 0 & & & a_{nn} \\ \end{matrix} \right\vert = \prod_{i=1}^{n}a_{ii}$$
2. 范氏行列式 $$V_{n} \triangleq \left\vert \begin{matrix} 1 & 1 & \cdots & 1 \\ a_{1} & a_{2} & \cdots & a_{n} \\ a^{2}_{1} & a^{2}_{2} & \cdots & a^{2}_{n} \\ \vdots & \vdots & \ddots & \vdots \\ a^{n-1}_{1} & a^{n-1}_{2} & \cdots & a^{n-1}_{n} \\ \end{matrix} \right\vert = \prod_{1\le j<i \le n}^{}(a_{i}-a_{j})$$
   • 注 $V_{n}\ne 0 \Leftrightarrow a_{1},\cdots,a_{n}\text{均为单值(即各不相等)}$
3. 分块行列式 $$\left\vert \begin{matrix} A & 0 \\ 0 & B \end{matrix} \right\vert =\left\vert \begin{matrix} A & * \\ 0 & B \end{matrix} \right\vert = \left\vert \begin{matrix} A & 0 \\ * & B \end{matrix} \right\vert = \vert A \vert \vert B \vert$$