# 如何确定行列式展开式中有效项的个数？

## 二、解析

### §2.1 不一般的解法

$$\begin{vmatrix} \boldsymbol{A} \end{vmatrix} = \begin{vmatrix} \mathbf{\textcolor{orangered}{a}} & \textit{0} & \mathbf{\textcolor{orangered}{b}} & \textit{0} \\ \textit{0} & \mathbf{\textcolor{orangered}{c}} & \textit{0} & \mathbf{\textcolor{orangered}{d}} \\ \mathbf{\textcolor{orangered}{e}} & \textit{0} & \mathbf{\textcolor{orangered}{f}} & \textit{0} \\ \textit{0} & \mathbf{\textcolor{orangered}{g}} & \textit{0} & \mathbf{\textcolor{orangered}{h}} \end{vmatrix}$$

$$\begin{vmatrix} \mathbf{\textcolor{orangered}{a_{11}}} & a_{12} & \mathbf{\textcolor{orangered}{a_{13}}} & a_{14} \\ a_{21} & \mathbf{\textcolor{orangered}{a_{22}}} & a_{23} & \mathbf{\textcolor{orangered}{a_{24}}} \\ \mathbf{\textcolor{orangered}{a_{31}}} & a_{32} & \mathbf{\textcolor{orangered}{a_{33}}} & a_{34} \\ a_{41} & \mathbf{\textcolor{orangered}{a_{42}}} & a_{43} & \mathbf{\textcolor{orangered}{a_{44}}} \\ \end{vmatrix}$$

\begin{aligned} \textcolor{springgreen}{a} & \Leftrightarrow \textcolor{springgreen}{a_{11}} \\ \textcolor{springgreen}{b} & \Leftrightarrow \textcolor{springgreen}{a_{13}} \\ \textcolor{yellow}{c} & \Leftrightarrow \textcolor{yellow}{a_{22}} \\ \textcolor{yellow}{d} & \Leftrightarrow \textcolor{yellow}{a_{24}} \\ \textcolor{orangered}{e} & \Leftrightarrow \textcolor{orangered}{a_{31}} \\ \textcolor{orangered}{f} & \Leftrightarrow \textcolor{orangered}{a_{33}} \\ \textcolor{pink}{g} & \Leftrightarrow \textcolor{pink}{a_{42}} \\ \textcolor{pink}{h} & \Leftrightarrow \textcolor{pink}{a_{44}} \end{aligned}

\begin{aligned} \begin{pmatrix} \textcolor{#161615}{\colorbox{#99F944}{1}} & & 3 & \\ & 2 & & \textcolor{#161615}{\colorbox{#99F944}{4}} \\ 1 & & \textcolor{#161615}{\colorbox{#99F944}{3}} & \\ & \textcolor{#161615}{\colorbox{#99F944}{2}} & & 4 \end{pmatrix} & \Rightarrow \textcolor{#161615}{\colorbox{#99F944}{1432}} \\ \\ \begin{pmatrix} \textcolor{#161615}{\colorbox{#99F944}{1}} & & 3 & \\ & \textcolor{#161615}{\colorbox{#99F944}{2}} & & 4 \\ 1 & & \textcolor{#161615}{\colorbox{#99F944}{3}} & \\ & 2 & & \textcolor{#161615}{\colorbox{#99F944}{4}} \end{pmatrix} & \Rightarrow \textcolor{#161615}{\colorbox{#99F944}{1234}} \\ \\ \begin{pmatrix} 1 & & \textcolor{#161615}{\colorbox{#99F944}{3}} & \\ & \textcolor{#161615}{\colorbox{#99F944}{2}} & & 4 \\ \textcolor{#161615}{\colorbox{#99F944}{1}} & & 3 & \\ & 2 & & \textcolor{#161615}{\colorbox{#99F944}{4}} \end{pmatrix} & \Rightarrow \textcolor{#161615}{\colorbox{#99F944}{3214}} \\ \\ \begin{pmatrix} 1 & & \textcolor{#161615}{\colorbox{#99F944}{3}} & \\ & 2 & & \textcolor{#161615}{\colorbox{#99F944}{4}} \\ \textcolor{#161615}{\colorbox{#99F944}{1}} & & 3 & \\ & \textcolor{#161615}{\colorbox{#99F944}{2}} & & 4 \end{pmatrix} & \Rightarrow \textcolor{#161615}{\colorbox{#99F944}{3412}} \\ \\ \end{aligned}

\begin{aligned} \begin{vmatrix} \boldsymbol{A} \end{vmatrix} \\ \\ & = (-1)^{\tau \left( \textcolor{#161615}{\colorbox{#99F944}{1432}} \right)} \cdot a_{1 \textcolor{#161615}{\colorbox{#99F944}{1}}} \cdot a_{2 \textcolor{#161615}{\colorbox{#99F944}{4}}} \cdot a_{3 \textcolor{#161615}{\colorbox{#99F944}{3}}} \cdot a_{4 \textcolor{#161615}{\colorbox{#99F944}{2}}} \\ & + (-1)^{\tau \left( \textcolor{#161615}{\colorbox{#99F944}{1234}} \right)} \cdot a_{1 \textcolor{#161615}{\colorbox{#99F944}{1}}} \cdot a_{2 \textcolor{#161615}{\colorbox{#99F944}{2}}} \cdot a_{3 \textcolor{#161615}{\colorbox{#99F944}{3}}} \cdot a_{4 \textcolor{#161615}{\colorbox{#99F944}{4}}} \\ & + (-1)^{\tau \left( \textcolor{#161615}{\colorbox{#99F944}{3214}} \right)} \cdot a_{1 \textcolor{#161615}{\colorbox{#99F944}{3}}} \cdot a_{2 \textcolor{#161615}{\colorbox{#99F944}{2}}} \cdot a_{3 \textcolor{#161615}{\colorbox{#99F944}{1}}} \cdot a_{4 \textcolor{#161615}{\colorbox{#99F944}{4}}} \\ & + (-1)^{\tau \left( \textcolor{#161615}{\colorbox{#99F944}{3412}} \right)} \cdot a_{1 \textcolor{#161615}{\colorbox{#99F944}{3}}} \cdot a_{2 \textcolor{#161615}{\colorbox{#99F944}{4}}} \cdot a_{3 \textcolor{#161615}{\colorbox{#99F944}{1}}} \cdot a_{4 \textcolor{#161615}{\colorbox{#99F944}{2}}} \\ \\ & = (-1)^{3} \cdot a_{1 \textcolor{#161615}{\colorbox{#99F944}{1}}} \cdot a_{2 \textcolor{#161615}{\colorbox{#99F944}{4}}} \cdot a_{3 \textcolor{#161615}{\colorbox{#99F944}{3}}} \cdot a_{4 \textcolor{#161615}{\colorbox{#99F944}{2}}} \\ & + (-1)^{0} \cdot a_{1 \textcolor{#161615}{\colorbox{#99F944}{1}}} \cdot a_{2 \textcolor{#161615}{\colorbox{#99F944}{2}}} \cdot a_{3 \textcolor{#161615}{\colorbox{#99F944}{3}}} \cdot a_{4 \textcolor{#161615}{\colorbox{#99F944}{4}}} \\ & + (-1)^{3} \cdot a_{1 \textcolor{#161615}{\colorbox{#99F944}{3}}} \cdot a_{2 \textcolor{#161615}{\colorbox{#99F944}{2}}} \cdot a_{3 \textcolor{#161615}{\colorbox{#99F944}{1}}} \cdot a_{4 \textcolor{#161615}{\colorbox{#99F944}{4}}} \\ & + (-1)^{4} \cdot a_{1 \textcolor{#161615}{\colorbox{#99F944}{3}}} \cdot a_{2 \textcolor{#161615}{\colorbox{#99F944}{4}}} \cdot a_{3 \textcolor{#161615}{\colorbox{#99F944}{1}}} \cdot a_{4 \textcolor{#161615}{\colorbox{#99F944}{2}}} \\ \\ & = -a_{1 1} \cdot a_{2 4} \cdot a_{3 3} \cdot a_{4 2} \\ & + a_{1 1} \cdot a_{2 2} \cdot a_{3 3} \cdot a_{4 4} \\ & – a_{1 3} \cdot a_{2 2} \cdot a_{3 1} \cdot a_{4 4} \\ & + a_{1 3} \cdot a_{2 4} \cdot a_{3 1} \cdot a_{4 2} \\ \\ & = a_{11} a_{33} \left( a_{22} a_{44} – a_{24} a_{42} \right) \\ & + a_{13} a_{31} \left( a_{24} a_{42} – a_{22} a_{44} \right) \\ \\ & = \left( a_{22} a_{44} – a_{24} a_{42} \right) \left( a_{11} a_{33} – a_{13} a_{31} \right) \\ \\ & = \textcolor{springgreen}{\boldsymbol{ \left( ch – dg \right) \left( af – be \right) }} \end{aligned}

\begin{aligned} \textcolor{springgreen}{a} & \Leftrightarrow \textcolor{springgreen}{a_{11}} & \textcolor{springgreen}{b} & \Leftrightarrow \textcolor{springgreen}{a_{13}} \\ \textcolor{yellow}{c} & \Leftrightarrow \textcolor{yellow}{a_{22}} & \textcolor{yellow}{d} & \Leftrightarrow \textcolor{yellow}{a_{24}} \\ \textcolor{orangered}{e} & \Leftrightarrow \textcolor{orangered}{a_{31}} & \textcolor{orangered}{f} & \Leftrightarrow \textcolor{orangered}{a_{33}} \\ \textcolor{pink}{g} & \Leftrightarrow \textcolor{pink}{a_{42}} & \textcolor{pink}{h} & \Leftrightarrow \textcolor{pink}{a_{44}} \end{aligned}

### §2.2 一般的解法

\begin{aligned} \begin{vmatrix} \boldsymbol{A} \end{vmatrix} \\ \\ & = \begin{vmatrix} a & 0 & b & 0 \\ 0 & c & 0 & d \\ e & 0 & f & 0 \\ 0 & g & 0 & h \end{vmatrix} \\ \\ & = \begin{vmatrix} \textcolor{black}{\colorbox{pink}{a}} & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{b}} & \textcolor{black}{\colorbox{pink}{0}} \\ \textcolor{black}{\colorbox{pink}{0}} & c & 0 & d \\ \textcolor{black}{\colorbox{pink}{e}} & 0 & f & 0 \\ \textcolor{black}{\colorbox{pink}{0}} & g & 0 & h \end{vmatrix} \\ \\ & = \begin{vmatrix} \textcolor{black}{\colorbox{pink}{a}} & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{b}} & \textcolor{black}{\colorbox{pink}{0}} \\ 0 & c & \textcolor{black}{\colorbox{pink}{0}} & d \\ e & 0 & \textcolor{black}{\colorbox{pink}{f}} & 0 \\ 0 & g & \textcolor{black}{\colorbox{pink}{0}} & h \end{vmatrix} \\ \\ & = (-1) ^{1+1} \cdot a \begin{vmatrix} c & 0 & d \\ 0 & f & 0 \\ g & 0 & h \end{vmatrix} + (-1) ^{1+3} \cdot b \begin{vmatrix} 0 & c & d \\ e & 0 & 0 \\ 0 & g & h \end{vmatrix} \\ \\ & = a \left[ cfh – dfg \right] + b \left[ dge – ceh \right] \\ \\ & = acfg – adfg + bdge – bceh \\ \\ & = afch – bech + bedg – afdg \\ \\ & = ch \left( af – be \right) + dg \left( be – af \right) \\ \\ & = \textcolor{springgreen}{\boldsymbol{ \left( ch – dg \right) \left( af – be \right) }} \end{aligned}

Next

\begin{aligned} \begin{vmatrix} \boldsymbol{A} \end{vmatrix} \\ \\ & = \begin{vmatrix} a & 0 & b & 0 \\ 0 & c & 0 & d \\ e & 0 & f & 0 \\ 0 & g & 0 & h \end{vmatrix} \\ \\ & = \begin{vmatrix} \textcolor{black}{\colorbox{pink}{a}} & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{b}} & \textcolor{black}{\colorbox{pink}{0}} \\ \textcolor{black}{\colorbox{pink}{0}} & c & 0 & d \\ \textcolor{black}{\colorbox{pink}{e}} & 0 & f & 0 \\ \textcolor{black}{\colorbox{pink}{0}} & g & 0 & h \end{vmatrix} \\ \\ & = \begin{vmatrix} \textcolor{black}{\colorbox{pink}{a}} & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{b}} & \textcolor{black}{\colorbox{pink}{0}} \\ 0 & c & \textcolor{black}{\colorbox{pink}{0}} & d \\ e & 0 & \textcolor{black}{\colorbox{pink}{f}} & 0 \\ 0 & g & \textcolor{black}{\colorbox{pink}{0}} & h \end{vmatrix} \\ \\ & = (-1) ^{1+1} \cdot a \begin{vmatrix} c & 0 & d \\ 0 & f & 0 \\ g & 0 & h \end{vmatrix} + (-1) ^{1+3} \cdot b \begin{vmatrix} 0 & c & d \\ e & 0 & 0 \\ 0 & g & h \end{vmatrix} \\ \\ & = a \begin{vmatrix} c & 0 & d \\ 0 & f & 0 \\ g & 0 & h \end{vmatrix} + b \begin{vmatrix} 0 & c & d \\ e & 0 & 0 \\ 0 & g & h \end{vmatrix} \\ \\ & = a \begin{vmatrix} \textcolor{white}{\colorbox{brown}{c}} & \textcolor{white}{\colorbox{brown}{0}} & \textcolor{white}{\colorbox{brown}{d}} \\ \textcolor{white}{\colorbox{brown}{0}} & f & 0 \\ \textcolor{white}{\colorbox{brown}{g}} & 0 & h \end{vmatrix} + b \begin{vmatrix} \textcolor{white}{\colorbox{brown}{0}} & \textcolor{white}{\colorbox{brown}{c}} & \textcolor{white}{\colorbox{brown}{d}} \\ e & \textcolor{white}{\colorbox{brown}{0}} & 0 \\ 0 & \textcolor{white}{\colorbox{brown}{g}} & h \end{vmatrix} \\ \\ & = a \begin{vmatrix} \textcolor{white}{\colorbox{brown}{c}} & \textcolor{white}{\colorbox{brown}{0}} & \textcolor{white}{\colorbox{brown}{d}} \\ 0 & f & \textcolor{white}{\colorbox{brown}{0}} \\ g & 0 & \textcolor{white}{\colorbox{brown}{h}} \end{vmatrix} + b \begin{vmatrix} \textcolor{white}{\colorbox{brown}{0}} & \textcolor{white}{\colorbox{brown}{c}} & \textcolor{white}{\colorbox{brown}{d}} \\ e & 0 & \textcolor{white}{\colorbox{brown}{0}} \\ 0 & g & \textcolor{white}{\colorbox{brown}{h}} \end{vmatrix} \\ \\ & = (-1) ^{1+1} ac \begin{vmatrix} f & 0 \\ 0 & h \end{vmatrix} + (-1) ^{1+3} ad \begin{vmatrix} 0 & f \\ g & 0 \end{vmatrix} \\ & + (-1)^{1+2} bc \begin{vmatrix} e & 0 \\ 0 & h \end{vmatrix} + (-1)^{1+3} bd \begin{vmatrix} e & 0 \\ 0 & g \end{vmatrix} \\ \\ & = acfh – adfg – bceh + bdeg \\ \\ & = afch – bech + bedg – afdg \\ \\ & = ch \left( af – be \right) + dg \left( be – af \right) \\ \\ & = \textcolor{springgreen}{\boldsymbol{ \left( ch – dg \right) \left( af – be \right) }} \end{aligned}