一.行列式
1.行列式定义
\(
\left | \begin{matrix}
a_{11} &a_{12} &... &a_{1n} \\
a_{21} &a_{22} &... &a_{2n} \\
... &... &... &...\\
a_{n1} &a_{n2} &... &a_{nn}
\end{matrix} \right |
\)
将此称为\(n\)阶行列式
设\(p\)数组为\(n\)的全排列,\(t\)为\(p\)数组的逆序对数
则此行列式的值为\(\sum (-1)^na_{1p_{1}}a_{1p_{2}}...a_{1p_{n}}\)
2.重要的结论及性质
1.上三角行列式的值为对角线的乘积
\[\left | \begin{matrix} a_{11} &a_{12} &... &a_{1n} \\ 0 &a_{22} &... &a_{2n} \\ ... &... &... &...\\ 0 &0 &... &a_{nn} \end{matrix} \right | = a_{11}a_{22}...a_{nn} \]2.下三角行列式的值为对角线的乘积
\[\left | \begin{matrix} a_{11} &0 &... &0 \\ a_{21} &a_{22} &... &0 \\ ... &... &... &...\\ a_{n1} &a_{n2} &... &a_{nn} \end{matrix} \right | = a_{11}a_{22}...a_{nn} \]3.对角行列式的值为对角线的乘积
\[\left | \begin{matrix} a_{11} &0 &... &0 \\ 0 &a_{22} &... &0 \\ ... &... &... &...\\ 0 &0 &... &a_{nn} \end{matrix} \right | = a_{11}a_{22}...a_{nn} \]4.若互换行列式的任意两行（列），则行列式变号
\[\left | \begin{matrix} a_{11} &a_{12} &... &a_{1n} \\ a_{21} &a_{22} &... &a_{2n} \\ ... &... &... &...\\ a_{n1} &a_{n2} &... &a_{nn} \end{matrix} \right | = -\left | \begin{matrix} a_{21} &a_{22} &... &a_{2n} \\ a_{11} &a_{12} &... &a_{1n} \\ ... &... &... &...\\ a_{n1} &a_{n2} &... &a_{nn} \end{matrix} \right | \]5.若行列式的一行乘上一个数k,则行列式的值也乘上一个数k
\[\left | \begin{matrix} a_{11} &a_{12} &... &a_{1n} \\ a_{21}*k &a_{22}*k &... &a_{2n}*k \\ ... &... &... &...\\ a_{n1} &a_{n2} &... &a_{nn} \end{matrix} \right | = -\left | \begin{matrix} a_{11} &a_{12} &... &a_{1n} \\ a_{21} &a_{22} &... &a_{2n} \\ ... &... &... &...\\ a_{n1} &a_{n2} &... &a_{nn} \end{matrix} \right | \]