本文是“线性代数”系列的第 01 章，主题为“行列式”。

下一篇：线性代数 Ch02：矩阵

\(n\)阶行列式的完全展开式
\[ \left| \begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22} & \cdots & a_{2n}\\ \vdots & \vdots & & \vdots\\ a_{n1} & a_{n2} & \cdots & a_{nn}\\ \end{array}\right| =\sum_{j_1j_2\cdots j_n}(-1)^{\tau(j_1j_2\cdots j_n)}a_{1j_1}a_{2j_2}\cdots a_{nj_n} \]

\(\tau(j_1j_2\cdots j_n)\)表示排列\(j_1j_2\cdots j_n\)的逆序数.

\(a_{ij}\)的余子式\(M_{ij}\): \(n\)阶行列式中划去第\(i\)行第\(j\)列剩下的元素构成的\(n-1\)阶行列式.

\(a_{ij}\)的代数余子式\(A_{ij}=(-1)^{i+j}M_{ij}\).

行列式按第\(k\)行(列)的展开公式: \(n\)阶行列式等于任意一行(列)的所有元素与各自对应的代数余子式的乘积之和.

副对角线的行列式
\[ \left| \begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22} & \cdots & 0\\ \vdots & \vdots & & \vdots\\ a_{n1} & 0 & \cdots & 0\\ \end{array}\right| =\left| \begin{array}{cccc} 0 & \cdots & 0 & a_{1n}\\ 0 & \cdots & a_{2,n-1} & a_{2n}\\ \vdots & & \vdots & \vdots\\ a_{n1} & \cdots & a_{n,n-1} & a_{nn}\\ \end{array}\right| =(-1)^{\frac{n(n-1)}{2}}a_{1n}a_{2,n-1}\cdots a_{n1} \]

拉普拉斯展开式
\[ \begin{aligned} &\left| \begin{array}{cc} \boldsymbol{A} & *\\ \boldsymbol{O} & \boldsymbol{B}\\ \end{array}\right| =\left| \begin{array}{cc} \boldsymbol{A} & \boldsymbol{O}\\ * & \boldsymbol{B}\\ \end{array}\right| =|\boldsymbol{A}|\cdot|\boldsymbol{B}|\\ &\left| \begin{array}{cc} \boldsymbol{O} & \boldsymbol{A}\\ \boldsymbol{B} & *\\ \end{array}\right| =\left| \begin{array}{cc} * & \boldsymbol{A}\\ \boldsymbol{B} & \boldsymbol{O}\\ \end{array}\right| =(-1)^{mn}|\boldsymbol{A}|\cdot|\boldsymbol{B}| \end{aligned} \]

范德蒙行列式
\[ \left| \begin{array}{cccc} 1 & 1 & \cdots & 1\\ x_1 & x_2 & \cdots & x_n\\ x_1^2 & x_2^2 & \cdots & x_n^2\\ \vdots & \vdots & & \vdots\\ x_1^{n-1} & x_2^{n-1} & \cdots & x_n^{n-1}\\ \end{array}\right| =\prod_{1\leqslant j<i\leqslant n}(x_i-x_j) \]

方阵的行列式

若\(\boldsymbol{A},\boldsymbol{B}\)为\(n\)阶矩阵,

\(|\boldsymbol{A}^T|=|\boldsymbol{A}|\) \(|k\boldsymbol{A}|=k^n|\boldsymbol{A}|\) \(|\boldsymbol{A}\boldsymbol{B}|=|\boldsymbol{A}|\cdot|\boldsymbol{B}|\) \(|\boldsymbol{A}^*|=|\boldsymbol{A}|^{n-1}\)

\(|\boldsymbol{A}^{-1}|=|\boldsymbol{A}|^{-1}\) \(|\boldsymbol{A}|=\displaystyle\prod_{i=1}^{n}\lambda_i\) \(\boldsymbol{A}\sim\boldsymbol{B}\Rightarrow|\boldsymbol{A}|=|\boldsymbol{B}|,|\boldsymbol{A}+k\boldsymbol{E}|=|\boldsymbol{B}+k\boldsymbol{E}|\)

注意\(|\boldsymbol{A}+\boldsymbol{B}|\)没有运算法则, 利用\(\boldsymbol{E}\)作恒等变形是常用技巧.

克拉默法则及其推论

\(n\)个方程\(n\)个未知数的线性方程组的系数行列式\(\boldsymbol{D}\neq0\Rightarrow\)方程组有唯一解.

推论:

(1) 齐次线性方程组系数行列式不为\(0\Rightarrow\)方程组只有零解.

(2) 齐次线性方程组有非零解\(\Rightarrow\)系数行列式为0.

矩阵秩的概念

\(r(\boldsymbol{A})=r\Leftrightarrow\boldsymbol{A}\)中有\(r\)阶子式不为0, 任何\(r+1\)阶子式(若还有)必全为0;

\(r(\boldsymbol{A})<r\Leftrightarrow\boldsymbol{A}\)中每一个\(r\)阶子式全为0;

\(r(\boldsymbol{A})\geqslant r\Leftrightarrow\boldsymbol{A}\)中有\(r\)阶子式不为0.

特别地, \(r(\boldsymbol{A})=0\Leftrightarrow\boldsymbol{A}=\boldsymbol{O}\); \(r(\boldsymbol{A})\geqslant1\Leftrightarrow\boldsymbol{A}\neq\boldsymbol{O}\).

若\(\boldsymbol{A}\)是\(n\)阶矩阵, \(r(\boldsymbol{A})=n\Leftrightarrow|\boldsymbol{A}|\neq0\Leftrightarrow\boldsymbol{A}\)可逆.

\(r(\boldsymbol{A})<n\Leftrightarrow|\boldsymbol{A}|=0\Leftrightarrow\boldsymbol{A}\)不可逆.

若\(A\)是\(m\times n\)矩阵, 则\(r(\boldsymbol{A})\leqslant\min(m,n)\).

设\(\boldsymbol{A}\)是\(n\)阶反对称矩阵, 即\(\boldsymbol{A}^T=-\boldsymbol{A}\), 若\(\boldsymbol{A}\)可逆, 则\(n\)必是偶数.

对于\(\boldsymbol{A}\boldsymbol{B}=\boldsymbol{O}\)的两种思路

设\(\boldsymbol{A}\)是\(m\times n\)矩阵, \(\boldsymbol{B}\)是\(n\times s\)矩阵, 若\(\boldsymbol{A}\boldsymbol{B}=\boldsymbol{O}\), 则

(1) \(\boldsymbol{B}\)的列向量是齐次方程组\(\boldsymbol{Ax}=\boldsymbol{0}\)的解;

(2) \(r(\boldsymbol{A})+r(\boldsymbol{B})\leqslant n\).

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下一篇：线性代数 Ch02：矩阵