标签： 计算数字型行列式的常用公式

$\left|\begin{array}{ccc}1& 1& 1\\ x_1& x_2& x_3\\ x_1^2& x_2^2& x_3^2\end{array}\right|$ $=$ $(x_2-x_1)$ $\cdot$ $(x_3-x_1)$ $\cdot$ $(x_3-x_2)$

范德蒙行列式的通用计算方式如下：

$\left|\begin{array}{ccccc}1& 1& 1& \cdots & 1\\ x_1& x_2& x_3& \cdots & x_n\\ x_1^2& x_2^2& x_3^2& \cdots & x_n^2\\ ⋮& ⋮& ⋮& & ⋮\\ x_1^{n-1}& x_2^{n-1}& x_3^{n-1}& \cdots & x_n^{n-1}\end{array}\right|$ $=$

$\prod _{1\le j<i\le n}$ $\left(x_i–x_j\right)$
总结：用右边的元素把左边的元素全减一遍并相乘。

$D_n$ $=$ $\left|\begin{array}{ccc}1& 1& 1\\ x_1& x_2& x_3\\ x_1^2& x_2^2& x_3^2\end{array}\right|$

范德蒙行列式的通用形式：

$\left|\begin{array}{ccccc}{x}_1^0& x_2^0& x_3^0& \cdots & x_n^0\\ x_1^1& x_2^1& x_3^1& \cdots & x_n^1\\ x_1^2& x_2^2& x_3^2& \cdots & x_n^2\\ ⋮& ⋮& ⋮& & ⋮\\ x_1^{n-1}& x_2^{n-1}& x_3^{n-1}& \cdots & x_n^{n-1}\end{array}\right|$ $\Rightarrow$

$\left|\begin{array}{ccccc}1& 1& 1& \cdots & 1\\ x_1& x_2& x_3& \cdots & x_n\\ x_1^2& x_2^2& x_3^2& \cdots & x_n^2\\ ⋮& ⋮& ⋮& & ⋮\\ x_1^{n-1}& x_2^{n-1}& x_3^{n-1}& \cdots & x_n^{n-1}\end{array}\right|$.

$\left|\begin{array}{cc}\mathit{O}& \mathit{A}\\ \mathit{B}& \mathit{C}\end{array}\right|$ $=$ $(-1{)}^{mn}$ $|\mathit{A}||\mathit{B}|$

$\left|\begin{array}{ll}\mathit{C}& \mathit{A}\\ \mathit{B}& \mathit{O}\end{array}\right|$ $=$ $(-1{)}^{mn}$ $|\mathit{A}||\mathit{B}|$

$\left|\begin{array}{ll}\mathit{O}& \mathit{A}\\ \mathit{B}& \mathit{O}\end{array}\right|$ $=$ $(-1{)}^{mn}$ $|\mathit{A}||\mathit{B}|$

$\left|\begin{array}{ll}\mathit{A}& \mathit{O}\\ \mathit{C}& \mathit{B}\end{array}\right|$ $=$ $|\mathit{A}||\mathit{B}|$

$\left|\begin{array}{ll}\mathit{A}& \mathit{C}\\ \mathit{O}& \mathit{B}\end{array}\right|$ $=$ $|\mathit{A}||\mathit{B}|$

$\left|\begin{array}{ll}\mathit{A}& \mathit{O}\\ \mathit{O}& \mathit{B}\end{array}\right|$ $=$ $|\mathit{A}||\mathit{B}|$

$\left|\begin{array}{cccc}0& & & {\lambda }_1\\ & & {\lambda }_2& \ast \\ & \cdots & ⋮& ⋮\\ {\lambda }_n& \cdots & \ast & \ast \end{array}\right|$ $=$ $(-1{)}^{\frac{n(n-1)}{2}}$ ${\lambda }_1$ ${\lambda }_2$ $\cdots$ ${\lambda }_n$

$\left|\begin{array}{cccc}\ast & \cdots & \ast & {\lambda }_1\\ \ast & \cdots & {\lambda }_2& \\ ⋮& \cdots & & \\ {\lambda }_n& & & 0\end{array}\right|$ $=$ $(-1{)}^{\frac{n(n-1)}{2}}$ ${\lambda }_1$ ${\lambda }_2$ $\cdots$ ${\lambda }_n$

$\left|\begin{array}{cccc}0& & & {\lambda }_1\\ & & {\lambda }_2& \\ & \cdots & & \\ {\lambda }_n& & & 0\end{array}\right|$ $=$ $(-1{)}^{\frac{n(n-1)}{2}}$ ${\lambda }_1$ ${\lambda }_2$ $\cdots$ ${\lambda }_n$

$\left|\begin{array}{ccc}\ast & \ast & \circ \\ \ast & \circ & \ast \\ \circ & \ast & \ast \end{array}\right|$

$\left|\begin{array}{ccc}\odot & \ast & \ast \\ \ast & \odot & \ast \\ \ast & \ast & \odot \end{array}\right|$

$\left|\begin{array}{cccc}{a}_{11}& 0& \cdots & 0\\ a_{21}& a_{22}& \cdots & 0\\ \cdots & \cdots & \cdots & \cdots \\ a_{n1}& a_{n2}& \cdots & a_{nn}\end{array}\right|$ $=$ $a_{11}$ $a_{22}$ $\cdots$ $a_{nn}$

$\left|\begin{array}{cccc}{a}_{11}& a_{12}& \cdots & a_{1n}\\ 0& a_{22}& \cdots & a_{2n}\\ \cdots & \cdots & \cdots & \cdots \\ 0& 0& \cdots & a_{nn}\end{array}\right|$ $=$ $a_{11}$ $a_{22}$ $\cdots$ $a_{nn}$