# 行列式“剥洋葱”：对于行或者列之间存在普遍规律的行列式可以尝试先提取其“公共部分”

## 一、题目

$$|V| = \begin{vmatrix} & a_{1}^{3} & a_{1}^{2}b_{1} & a_{1}b_{1}^{2} & b_{1}^{3} & \\ \\ & a_{2}^{3} & a_{2}^{2}b_{2} & a_{2}b_{2}^{2} & b_{2}^{3} & \\ \\ & a_{3}^{3} & a_{3}^{2}b_{3} & a_{3}b_{3}^{2} & b_{3}^{3} & \\ \\ & a_{4}^{3} & a_{4}^{2}b_{4} & a_{4}b_{4}^{2} & b_{4}^{3} & \end{vmatrix} ⁢= ?$$

## 二、解析

$$a_{1} ^{2} \quad a_{1} b_{1} \quad b_{1} ^{2} \quad \frac{b_{1} ^{3}}{a_{1}}$$

$$a_{1} \quad b_{1} \quad \frac{b_{1} ^{2}}{a_{1}} \quad \frac{b_{1} ^{3}}{a_{1} ^{2}}$$

$$1 \quad \frac{b_{1}}{a_{1}} \quad \frac{b_{1} ^{2}}{a_{1} ^{2}} \quad \frac{b_{1} ^{3}}{a_{1} ^{3}}$$

\begin{aligned} & \begin{vmatrix} & a_{1}^{3} & a_{1}^{2}b_{1} & a_{1}b_{1}^{2} & b_{1}^{3} & \\ \\ & a_{2}^{3} & a_{2}^{2}b_{2} & a_{2}b_{2}^{2} & b_{2}^{3} & \\ \\ & a_{3}^{3} & a_{3}^{2}b_{3} & a_{3}b_{3}^{2} & b_{3}^{3} & \\ \\ & a_{4}^{3} & a_{4}^{2}b_{4} & a_{4}b_{4}^{2} & b_{4}^{3} & \end{vmatrix} \\ \\ \textcolor{pink}{ \Rightarrow } a_{1} ^{3} & \begin{vmatrix} & 1 & \frac{b_{1}}{a_{1}} & \frac{b_{1}^{2}}{a_{1} ^{2}} & \frac{b_{1}^{3}}{a_{1} ^{3}} & \\ \\ & a_{2}^{3} & a_{2}^{2}b_{2} & a_{2}b_{2}^{2} & b_{2}^{3} & \\ \\ & a_{3}^{3} & a_{3}^{2}b_{3} & a_{3}b_{3}^{2} & b_{3}^{3} & \\ \\ & a_{4}^{3} & a_{4}^{2}b_{4} & a_{4}b_{4}^{2} & b_{4}^{3} & \end{vmatrix} \\ \\ \textcolor{pink}{ \Rightarrow } a_{1} ^{3} a_{2} ^{3} & \begin{vmatrix} & 1 & \frac{b_{1}}{a_{1}} & \frac{b_{1}^{2}}{a_{1} ^{2}} & \frac{b_{1}^{3}}{a_{1} ^{3}} & \\ \\ & 1 & \frac{b_{2}}{a_{2}} & \frac{b_{2}^{2}}{a_{2} ^{2}} & \frac{b_{2}^{3}}{a_{2} ^{3}} & \\ \\ & a_{3}^{3} & a_{3}^{2}b_{3} & a_{3}b_{3}^{2} & b_{3}^{3} & \\ \\ & a_{4}^{3} & a_{4}^{2}b_{4} & a_{4}b_{4}^{2} & b_{4}^{3} & \end{vmatrix} \\ \\ \textcolor{pink}{ \Rightarrow } a_{1} ^{3} a_{2} ^{3} a_{3} ^{3} & \begin{vmatrix} & 1 & \frac{b_{1}}{a_{1}} & \frac{b_{1}^{2}}{a_{1} ^{2}} & \frac{b_{1}^{3}}{a_{1} ^{3}} & \\ \\ & 1 & \frac{b_{2}}{a_{2}} & \frac{b_{2}^{2}}{a_{2} ^{2}} & \frac{b_{2}^{3}}{a_{2} ^{3}} & \\ \\ & 1 & \frac{b_{3}}{a_{3}} & \frac{b_{3}^{2}}{a_{3} ^{2}} & \frac{b_{3}^{3}}{a_{3} ^{3}} & \\ \\ & a_{4}^{3} & a_{4}^{2}b_{4} & a_{4}b_{4}^{2} & b_{4}^{3} & \end{vmatrix} \\ \\ \textcolor{pink}{ \Rightarrow } a_{1} ^{3} a_{2} ^{3} a_{3} ^{3} a_{4} ^{3} & \begin{vmatrix} & 1 & \frac{b_{1}}{a_{1}} & \frac{b_{1}^{2}}{a_{1} ^{2}} & \frac{b_{1}^{3}}{a_{1} ^{3}} & \\ \\ & 1 & \frac{b_{2}}{a_{2}} & \frac{b_{2}^{2}}{a_{2} ^{2}} & \frac{b_{2}^{3}}{a_{2} ^{3}} & \\ \\ & 1 & \frac{b_{3}}{a_{3}} & \frac{b_{3}^{2}}{a_{3} ^{2}} & \frac{b_{3}^{3}}{a_{3} ^{3}} & \\ \\ & 1 & \frac{b_{4}}{a_{4}} & \frac{b_{4}^{2}}{a_{4} ^{2}} & \frac{b_{4}^{3}}{a_{4} ^{3}} & \end{vmatrix} \\ \\ \textcolor{pink}{ \Rightarrow } a_{1} ^{3} a_{2} ^{3} a_{3} ^{3} a_{4} ^{3} & \begin{vmatrix} & \textcolor{orangered}{1} & \frac{b_{1}}{a_{1}} & \frac{b_{1}^{2}}{a_{1} ^{2}} & \frac{b_{1}^{3}}{a_{1} ^{3}} & \\ \\ & 0 & \frac{b_{2}}{a_{2}} – \frac{b_{1}}{a_{1}} & \frac{b_{2}^{2}}{a_{2} ^{2}} – \frac{b_{1}^{2}}{a_{1} ^{2}} & \frac{b_{2}^{3}}{a_{2} ^{3}} – \frac{b_{1}^{3}}{a_{1} ^{3}} & \\ \\ & 0 & \frac{b_{3}}{a_{3}} – \frac{b_{1}}{a_{1}} & \frac{b_{3}^{2}}{a_{3} ^{2}} – \frac{b_{1}^{2}}{a_{1} ^{2}} & \frac{b_{3}^{3}}{a_{3} ^{3}} – \frac{b_{1}^{3}}{a_{1} ^{3}} & \\ \\ & 0 & \frac{b_{4}}{a_{4}} – \frac{b_{1}}{a_{1}} & \frac{b_{4}^{2}}{a_{4} ^{2}} – \frac{b_{1}^{2}}{a_{1} ^{2}} & \frac{b_{4}^{3}}{a_{4} ^{3}} – \frac{b_{1}^{3}}{a_{1} ^{3}} & \end{vmatrix} \\ \\ \textcolor{pink}{ \Rightarrow } a_{1} ^{3} a_{2} ^{3} a_{3} ^{3} a_{4} ^{3} \cdot \textcolor{orangered}{1} & \textcolor{springgreen}{ \begin{vmatrix} & \frac{b_{2}}{a_{2}} – \frac{b_{1}}{a_{1}} & \frac{b_{2}^{2}}{a_{2} ^{2}} – \frac{b_{1}^{2}}{a_{1} ^{2}} & \frac{b_{2}^{3}}{a_{2} ^{3}} – \frac{b_{1}^{3}}{a_{1} ^{3}} & \\ \\ & \frac{b_{3}}{a_{3}} – \frac{b_{1}}{a_{1}} & \frac{b_{3}^{2}}{a_{3} ^{2}} – \frac{b_{1}^{2}}{a_{1} ^{2}} & \frac{b_{3}^{3}}{a_{3} ^{3}} – \frac{b_{1}^{3}}{a_{1} ^{3}} & \\ \\ & \frac{b_{4}}{a_{4}} – \frac{b_{1}}{a_{1}} & \frac{b_{4}^{2}}{a_{4} ^{2}} – \frac{b_{1}^{2}}{a_{1} ^{2}} & \frac{b_{4}^{3}}{a_{4} ^{3}} – \frac{b_{1}^{3}}{a_{1} ^{3}} & \end{vmatrix} } \end{aligned}

\begin{aligned} & a_{1} ^{3} a_{2} ^{3} a_{3} ^{3} a_{4} ^{3} \textcolor{springgreen}{ \begin{vmatrix} & \frac{b_{2}}{a_{2}} – \frac{b_{1}}{a_{1}} & \frac{b_{2}^{2}}{a_{2} ^{2}} – \frac{b_{1}^{2}}{a_{1} ^{2}} & \frac{b_{2}^{3}}{a_{2} ^{3}} – \frac{b_{1}^{3}}{a_{1} ^{3}} & \\ \\ & \frac{b_{3}}{a_{3}} – \frac{b_{1}}{a_{1}} & \frac{b_{3}^{2}}{a_{3} ^{2}} – \frac{b_{1}^{2}}{a_{1} ^{2}} & \frac{b_{3}^{3}}{a_{3} ^{3}} – \frac{b_{1}^{3}}{a_{1} ^{3}} & \\ \\ & \frac{b_{4}}{a_{4}} – \frac{b_{1}}{a_{1}} & \frac{b_{4}^{2}}{a_{4} ^{2}} – \frac{b_{1}^{2}}{a_{1} ^{2}} & \frac{b_{4}^{3}}{a_{4} ^{3}} – \frac{b_{1}^{3}}{a_{1} ^{3}} & \end{vmatrix} } \\ \\ \\ \textcolor{pink}{ \Rightarrow } & a_{1} ^{3} a_{2} ^{3} a_{3} ^{3} a_{4} ^{3} \left( \frac{b_{2}}{a_{2}} – \frac{b_{1}}{a_{1}} \right) \times \\ & \begin{vmatrix} & 1 & \frac{b_{2}}{a_{2}} – \frac{b_{1}}{a_{1}} & \frac{b_{2}^{2}}{a_{2} ^{2}} – \frac{b_{1}^{2}}{a_{1} ^{2}} & \\ \\ & \frac{b_{3}}{a_{3}} – \frac{b_{1}}{a_{1}} & \frac{b_{3}^{2}}{a_{3} ^{2}} – \frac{b_{1}^{2}}{a_{1} ^{2}} & \frac{b_{3}^{3}}{a_{3} ^{3}} – \frac{b_{1}^{3}}{a_{1} ^{3}} & \\ \\ & \frac{b_{4}}{a_{4}} – \frac{b_{1}}{a_{1}} & \frac{b_{4}^{2}}{a_{4} ^{2}} – \frac{b_{1}^{2}}{a_{1} ^{2}} & \frac{b_{4}^{3}}{a_{4} ^{3}} – \frac{b_{1}^{3}}{a_{1} ^{3}} & \end{vmatrix} \\ \\ \\ \textcolor{pink}{ \Rightarrow } & a_{1} ^{3} a_{2} ^{3} a_{3} ^{3} a_{4} ^{3} \left( \frac{b_{2}}{a_{2}} – \frac{b_{1}}{a_{1}} \right) \left( \frac{b_{3}}{a_{3}} – \frac{b_{1}}{a_{1}} \right) \times \\ & \begin{vmatrix} & 1 & \frac{b_{2}}{a_{2}} – \frac{b_{1}}{a_{1}} & \frac{b_{2}^{2}}{a_{2} ^{2}} – \frac{b_{1}^{2}}{a_{1} ^{2}} & \\ \\ & 1 & \frac{b_{3}}{a_{3}} – \frac{b_{1}}{a_{1}} & \frac{b_{3}^{2}}{a_{3} ^{2}} – \frac{b_{1}^{2}}{a_{1} ^{2}} & \\ \\ & \frac{b_{4}}{a_{4}} – \frac{b_{1}}{a_{1}} & \frac{b_{4}^{2}}{a_{4} ^{2}} – \frac{b_{1}^{2}}{a_{1} ^{2}} & \frac{b_{4}^{3}}{a_{4} ^{3}} – \frac{b_{1}^{3}}{a_{1} ^{3}} & \end{vmatrix} \\ \\ \\ \textcolor{pink}{ \Rightarrow } & a_{1} ^{3} a_{2} ^{3} a_{3} ^{3} a_{4} ^{3} \left( \frac{b_{2}}{a_{2}} – \frac{b_{1}}{a_{1}} \right) \left( \frac{b_{3}}{a_{3}} – \frac{b_{1}}{a_{1}} \right) \left( \frac{b_{4}}{a_{4}} – \frac{b_{1}}{a_{1}} \right) \times \\ & \begin{vmatrix} & 1 & \frac{b_{2}}{a_{2}} – \frac{b_{1}}{a_{1}} & \frac{b_{2}^{2}}{a_{2} ^{2}} – \frac{b_{1}^{2}}{a_{1} ^{2}} & \\ \\ & 1 & \frac{b_{3}}{a_{3}} – \frac{b_{1}}{a_{1}} & \frac{b_{3}^{2}}{a_{3} ^{2}} – \frac{b_{1}^{2}}{a_{1} ^{2}} & \\ \\ & 1 & \frac{b_{4}}{a_{4} } – \frac{b_{1}}{a_{1} } & \frac{b_{4}^{2}}{a_{4} ^{2}} – \frac{b_{1}^{2}}{a_{1} ^{2}} & \end{vmatrix} \\ \\ \\ \textcolor{pink}{ \Rightarrow } & a_{1} ^{3} a_{2} ^{3} a_{3} ^{3} a_{4} ^{3} \left( \frac{b_{2}}{a_{2}} – \frac{b_{1}}{a_{1}} \right) \left( \frac{b_{3}}{a_{3}} – \frac{b_{1}}{a_{1}} \right) \left( \frac{b_{4}}{a_{4}} – \frac{b_{1}}{a_{1}} \right) \times \\ & \begin{vmatrix} & \textcolor{orangered}{1} & \frac{b_{2}}{a_{2}} – \frac{b_{1}}{a_{1}} & \frac{b_{2}^{2}}{a_{2} ^{2}} – \frac{b_{1}^{2}}{a_{1} ^{2}} & \\ \\ & 0 & \frac{b_{3}}{a_{3}} – \frac{b_{2}}{a_{2}} & \frac{b_{3}^{2}}{a_{3} ^{2}} – \frac{b_{2}^{2}}{a_{2} ^{2}} & \\ \\ & 0 & \frac{b_{4}}{a_{4} } – \frac{b_{2}}{a_{2} } & \frac{b_{4}^{2}}{a_{4} ^{2}} – \frac{b_{2}^{2}}{a_{2} ^{2}} & \end{vmatrix} \\ \\ \\ \textcolor{pink}{ \Rightarrow } & a_{1} ^{3} a_{2} ^{3} a_{3} ^{3} a_{4} ^{3} \left( \frac{b_{2}}{a_{2}} – \frac{b_{1}}{a_{1}} \right) \left( \frac{b_{3}}{a_{3}} – \frac{b_{1}}{a_{1}} \right) \left( \frac{b_{4}}{a_{4}} – \frac{b_{1}}{a_{1}} \right) \cdot \textcolor{orangered}{1} \times \\ & \begin{vmatrix} & \frac{b_{3}}{a_{3}} – \frac{b_{2}}{a_{2}} & \frac{b_{3}^{2}}{a_{3} ^{2}} – \frac{b_{2}^{2}}{a_{2} ^{2}} & \\ \\ & \frac{b_{4}}{a_{4} } – \frac{b_{2}}{a_{2} } & \frac{b_{4}^{2}}{a_{4} ^{2}} – \frac{b_{2}^{2}}{a_{2} ^{2}} & \end{vmatrix} \\ \\ \\ \textcolor{pink}{ \Rightarrow } & a_{1} ^{3} a_{2} ^{3} a_{3} ^{3} a_{4} ^{3} \left( \frac{b_{2}}{a_{2}} – \frac{b_{1}}{a_{1}} \right) \left( \frac{b_{3}}{a_{3}} – \frac{b_{1}}{a_{1}} \right) \left( \frac{b_{4}}{a_{4}} – \frac{b_{1}}{a_{1}} \right) \times \\ & \begin{vmatrix} & \frac{b_{3}}{a_{3}} – \frac{b_{2}}{a_{2}} & \frac{b_{3}^{2}}{a_{3} ^{2}} – \frac{b_{2}^{2}}{a_{2} ^{2}} & \\ \\ & \frac{b_{4}}{a_{4} } – \frac{b_{2}}{a_{2} } & \frac{b_{4}^{2}}{a_{4} ^{2}} – \frac{b_{2}^{2}}{a_{2} ^{2}} & \end{vmatrix} \\ \\ \\ \textcolor{pink}{ \Rightarrow } & a_{1} ^{3} a_{2} ^{3} a_{3} ^{3} a_{4} ^{3} \left( \frac{b_{2}}{a_{2}} – \frac{b_{1}}{a_{1}} \right) \left( \frac{b_{3}}{a_{3}} – \frac{b_{1}}{a_{1}} \right) \left( \frac{b_{4}}{a_{4}} – \frac{b_{1}}{a_{1}} \right) \left( \frac{b_{3}}{a_{3}} – \frac{b_{2}}{a_{2}} \right) \left( \frac{b_{4}}{a_{4} } – \frac{b_{2}}{a_{2} } \right) \times \\ & \begin{vmatrix} & 1 & \frac{b_{3}}{a_{3}} – \frac{b_{2}}{a_{2}} & \\ \\ & 1 & \frac{b_{4}}{a_{4}} – \frac{b_{2}}{a_{2}} & \end{vmatrix} \\ \\ \\ \textcolor{pink}{ \Rightarrow } & a_{1} ^{3} a_{2} ^{3} a_{3} ^{3} a_{4} ^{3} \left( \frac{b_{2}}{a_{2}} – \frac{b_{1}}{a_{1}} \right) \left( \frac{b_{3}}{a_{3}} – \frac{b_{1}}{a_{1}} \right) \left( \frac{b_{4}}{a_{4}} – \frac{b_{1}}{a_{1}} \right) \left( \frac{b_{3}}{a_{3}} – \frac{b_{2}}{a_{2}} \right) \left( \frac{b_{4}}{a_{4} } – \frac{b_{2}}{a_{2} } \right) \times \\ & \begin{vmatrix} & 1 & \frac{b_{3}}{a_{3}} – \frac{b_{2}}{a_{2}} & \\ \\ & 0 & \frac{b_{4}}{a_{4}} – \frac{b_{3}}{a_{3}} & \end{vmatrix} \\ \\ \\ \textcolor{pink}{ \Rightarrow } & \textcolor{green}{\boldsymbol{ \left( a_{1} a_{2} a_{3} a_{4} \right) ^{3} \left( \frac{b_{2}}{a_{2}} – \frac{b_{1}}{a_{1}} \right) \left( \frac{b_{3}}{a_{3}} – \frac{b_{1}}{a_{1}} \right) \left( \frac{b_{4}}{a_{4}} – \frac{b_{1}}{a_{1}} \right) \left( \frac{b_{3}}{a_{3}} – \frac{b_{2}}{a_{2}} \right) \left( \frac{b_{4}}{a_{4} } – \frac{b_{2}}{a_{2} } \right) \left( \frac{b_{4}}{a_{4}} – \frac{b_{3}}{a_{3}} \right) }} \\ \\ \\ \textcolor{pink}{ \Rightarrow } & \left( a _ { 1 } a _ { 2 } a _ { 3 } a _ { 4 } \right) ^ { 3 } \prod _ { 1 \leqslant j < i \leqslant 4 } \left( \frac { b _ { i } } { a _ { i } } – \frac { b _ { j } } { a _ { j } } \right) \end{aligned}