行列式
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行列式的定义
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记$S_n$为$1$到$n$的全排列，$N(k_1,k_2,\cdots,k_n)$为排列$(k_1,k_2,\cdots,k_n)$的逆序数，则矩阵$\boldsymbol{A}$的行列式$|\boldsymbol{A}|$定义为
$$|\boldsymbol{A}|=\sum\limits_{(k_1,k_2,\cdots,k_n)\in S_n}(-1)^{N(k_1,k_2,\cdots,k_n)}a_{k_1 1}a_{k_2 2}\cdots a_{k_3 3}$$
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行列式的性质
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$$1.两行互换行列式值反号\\
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2.任意一行乘常数c行列式值乘常数c\\
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3.任意一行乘常数c加到另一行行列式值不变$$
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行列式的计算
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$$|\boldsymbol{A}|=\sum\limits_{1\leq j_1<j_2<\cdots<j_k\leq n}\boldsymbol{A}\left(\begin{array}{cc}
i_1 &i_2 &\cdots &i_k\\
j_1 &j_2 &\cdots &j_k \end{array}\right)\widehat{\boldsymbol{A}}\left(\begin{array}{cc}
i_1 &i_2 &\cdots &i_k\\
j_1 &j_2 &\cdots &j_k \end{array}\right)\\
\quad\\
\widehat{\boldsymbol{A}}表示代数余子式$$
$$V_n=\left|\begin{array} &1&x_1&x_1^2&\cdots&x_1^{n-1}\\
1&x_2&x_2^2&\cdots&x_2^{n-1}\\
\vdots&\vdots&\vdots&&\vdots\\
1&x_{n-1}&x_{n-1}^2&\cdots&x_{n-1}^{n-1}\\
1&x_n&x_n^2&\cdots&x_n^{n-1} \end{array}\right|=\prod\limits_{1\leq i< j\leq n}^n (x_j-x_i)$$
$$|\boldsymbol{A}||\boldsymbol{D}-\boldsymbol{B}\boldsymbol{A}^{-1}\boldsymbol{C}|=|\boldsymbol{D}||\boldsymbol{A}-\boldsymbol{C}\boldsymbol{D}^{-1}\boldsymbol{B}|$$
设$\boldsymbol{A}$是$n$阶可逆阵，$\boldsymbol{\alpha}$，$\boldsymbol{\beta}$是$n$维列向量，且$1+\boldsymbol{\beta}^{T}\boldsymbol{A}^{-1}\boldsymbol{\alpha}\neq 0$，则有
$$(\boldsymbol{A}+\boldsymbol{\alpha}\boldsymbol{\beta}^T)^{-1}=\boldsymbol{A}^{-1}-\frac{\boldsymbol{A}^{-1}\boldsymbol{\alpha}\boldsymbol{\beta}^T\boldsymbol{A}^{-1}}{1+\boldsymbol{\beta}^T\boldsymbol{A}^{-1}\boldsymbol{\alpha}}$$