# 高阶行列式的计算思路：降阶或者找规律

## 二、解析

### 解法一

\begin{aligned} \left| \begin{array} { c c c c } a _{ 1 } & 0 & 0 & b _{ 1 } \\ 0 & a _{ 2 } & b _{ 2 } & 0 \\ 0 & b _{ 3 } & a _{ 3 } & 0 \\ b _{ 4 } & 0 & 0 & a _{ 4 } \end{array} \right| \\ \\ & \underrightarrow{\quad \textcolor{blue}{降为 \ 3 \ 阶行列式} \quad} \\ \\ & = a _{ 1 } \times ( – 1 ) ^ { 1 + 1 } \left| \begin{array} { c c c } a _{ 2 } & b _{ 2 } & 0 \\ b _{ 3 } & a _{ 3 } & 0 \\ 0 & 0 & a _{ 4 } \end{array} \right| \\ \\ & + b _{ 1 } \times ( – 1 ) ^ { 1 + 4 } \left| \begin{array} { c c c } 0 & a _{ 2 } & b _{ 2 } \\ 0 & b _{ 3 } & a _{ 3 } \\ b _{ 4 } & 0 & 0 \end{array} \right| \\ \\ & \underrightarrow{\quad \textcolor{blue}{降为 \ 2 \ 阶行列式} \quad} \\ \\ & = a _{ 1 } a _{ 4 } \left| \begin{array} { l l } a _{ 2 } & b _{ 2 } \\ b _{ 3 } & a _{ 3 } \end{array} \right| – b _{ 1 } b _{ 4 } \left| \begin{array} { l l } a _{ 2 } & b _{ 2 } \\ b _{ 3 } & a _{ 3 } \end{array} \right|\\ \\ & = \textcolor{springgreen}{\boldsymbol{\left( a _{ 2 } a _{ 3 } – b _{ 2 } b _{ 3 } \right) \left( a _{ 1 } a _{ 4 } – b _{ 1 } b _{ 4 } \right)}} \end{aligned}

### 解法二

\begin{aligned} & \left| \begin{array} { c c c c } a _{ 1 } & 0 & 0 & b _{ 1 } \\ 0 & a _{ 2 } & b _{ 2 } & 0 \\ 0 & b _{ 3 } & a _{ 3 } & 0 \\ b _{ 4 } & 0 & 0 & a _{ 4 } \end{array} \right| \underrightarrow{\quad 交换第 \ 2 \ 列和第 \ 4 \ 列 \quad } \\ \\ & \left| \begin{array} { c c c c } a _{ 1 } & b_{1} & 0 & 0 \\ 0 & 0 & b _{ 2 } & a_{2} \\ 0 & 0 & a _{ 3 } & b_{3}\\ b _{ 4 } & a_{4} & 0 & 0 \end{array} \right| \underrightarrow{\quad 交换第 \ 2 \ 行和第 \ 4 \ 行 \quad } \\ \\ & \left| \begin{array} { c c c c } \textcolor{springgreen}{a _{ 1 }} & \textcolor{springgreen}{b_{1}} & 0 & 0 \\ \textcolor{springgreen}{b _{ 4 }} & \textcolor{springgreen}{a_{4}} & 0 & 0 \\ 0 & 0 & \textcolor{orangered}{a _{ 3 }} & \textcolor{orangered}{b_{3}} \\ 0 & 0 & \textcolor{orangered}{b _{ 2 }} & \textcolor{orangered}{a_{2}} \end{array} \right| \\ \\ & = \begin{vmatrix} \textcolor{springgreen}{a_{1}} & \textcolor{springgreen}{b_{1}} \\ \textcolor{springgreen}{b_{4}} & \textcolor{springgreen}{a_{4}} \end{vmatrix} \cdot \begin{vmatrix} \textcolor{orangered}{a_{3}} & \textcolor{orangered}{b_{3}} \\ \textcolor{orangered}{b_{2}} & \textcolor{orangered}{a_{2}} \end{vmatrix} \\ \\ & = (a_{1} a_{4} – b_{1} b_{4}) (a_{2} a_{3} – b_{2} b_{3}) \\ \\ & = \textcolor{springgreen}{\boldsymbol{(a_{2} a_{3} – b_{2} b_{3}) (a_{1} a_{4} – b_{1} b_{4})}} \end{aligned}