行列式定义

2021-07-12  本文已影响0人  我是聪

n阶行列式

n阶行列式.png

n阶行列式

标识形式

\begin{vmatrix} a_{1,1} & a_{1,2} & a_{1,3}\\ a_{2,1} & a_{2,3} & a_{2,3} \\ a_{3,1} & a_{3,2} & a_{3,3} \end{vmatrix} = a_{1,1}a_{2,2}a_{3,3}+a_{1,2}a_{2,3}a_{3,1}+a_{1,3}a_{2,1}a_{3,2}-a_{1,3}a_{2,2}a_{3,1}-a_{1,2}a_{2,1}a_{3,3}-a_{1,1}a_{2,3}a_{3,2}

\begin{vmatrix} a_{1,1} & a_{1,2} & ... &a_{1,n}\\ a_{2,1} & a_{2,3} & ... &a_{2,n} \\ ... & ... & ... & ...\\ a_{n,1} & a_{n,2} & ... &a_{n,n} \end{vmatrix}=\sum_{j=1}^N (-1)^N(j_{1}j_{2}..j_{n})a_{1,j_{1}}a_{2,j_{2}}...a_{n,j_{n}}

        -

\begin{vmatrix} a_{i,j} \\ \end{vmatrix} = D
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\begin{vmatrix} a_{1,1}\\ \end{vmatrix} = a_{1,1}

- 按列展开

\begin{vmatrix} a_{1,1} & a_{1,2} & ... &a_{1,n}\\ a_{2,1} & a_{2,3} & ... &a_{2,n} \\ ... & ... & ... & ...\\ a_{n,1} & a_{n,2} & ... &a_{n,n} \end{vmatrix}=\sum_{i=1}^N (-1)^N(i_{1}i_{2}..i_{n})a_{i_{1},1}a_{i_{2},2}...a_{i_{n},n}

\begin{vmatrix} a_{1,1} & 0 & ... & 0 & 0 \\ a_{2,1} & a_{2,2} & ... & 0 & 0 \\ a_{3,1} & a_{3,4} & a_{3,3} & ... & 0\\ ... & ... & ... & ... & ...\\ a_{n,1} & a_{n,2} & a_{n,3} & ... & a_{n,n}\\ \end{vmatrix} = a_{1,1}a_{2,2}...a_{n,n}

        - 结果是主对角线元素相乘

- 上三角行列式

    -

\begin{vmatrix} a_{1,1} &a_{1,2}& ... & a_{1,n-1} & a_{1,n} \\ 0 & a_{2,2} & ... & a_{2,n-1} & a_{2,n} \\ 0 & 0 & a_{3,3} & ... & a_{3,n}\\ ... & ... & ... & ... & ...\\ 0 & 0 & 0 & ... & a_{n,n}\\ \end{vmatrix} = a_{1,1}a_{2,2}...a_{n,n}

        - 也过也是主对角线元素相乘

- 反下三角行列式

    -

\begin{vmatrix} 0 & ... & 0 & 0 &a_{1,n} \\ 0 & ... & 0 &a_{2,n-1} & a_{2,n} &\\ 0 & ... & a_{3,n-2} & a_{3,n-1} & a_{3,n} \\ ... & ... & ... & ... & ...\\ a_{n,1} & a_{n,2} & a_{n,3} & ... & a_{n,n}\\ \end{vmatrix} = (-1)^\tfrac{n(n-1)}{2} a_{1,n}a_{2,n-1}...a_{n,1}

- 反上三角行列式

    -

\begin{vmatrix} a_{1,1} & ... & a_{1,n-2} & a_{1,n-1} &a_{1,n} \\ a_{1,2} & ... & a_{1,n-2} & a_{2,n-1} & 0 \\ a_{1,3} & ... & a_{3,n-2} & 0 & 0 \\ ... & ... & ... & ... & ...\\ a_{n,1} & 0 & 0 & ... & 0\\ \end{vmatrix} = (-1)^\tfrac{n(n-1)}{2} a_{1,n}a_{2,n-1}...a_{n,1}

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