行列式中的“消消乐”

二、解析

$$\begin{vmatrix} 1 & -2 & 5 & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{0}} \\ 3 & 8 & 1 & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{0}} \\ 5 & 0 & -3 & 2 & 1 & -1 \\ 1 & 2 & 5 & 2 & 1 & -1 \\ 7 & 3 & 5 & 9 & 2 & 0 \\ 1 & 6 & 5 & -5 & 3 & 2 \\ \end{vmatrix}$$

$$\begin{vmatrix} 1 & -2 & 5 & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{0}} \\ 3 & 8 & 1 & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{0}} \\ 5 & 0 & -3 & \textcolor{black}{\colorbox{tan}{2}} & \textcolor{black}{\colorbox{tan}{1}} & \textcolor{black}{\colorbox{tan}{-1}} \\ 1 & 2 & 5 & \textcolor{black}{\colorbox{tan}{2}} & \textcolor{black}{\colorbox{tan}{1}} & \textcolor{black}{\colorbox{tan}{-1}} \\ 7 & 3 & 5 & 9 & 2 & 0 \\ 1 & 6 & 5 & -5 & 3 & 2 \\ \end{vmatrix}$$

\begin{aligned} & \begin{vmatrix} 1 & -2 & 5 & 0 & 0 & 0 \\ 3 & 8 & 1 & 0 & 0 & 0 \\ \textcolor{orangered}{5} & \textcolor{orangered}{0} & \textcolor{orangered}{-3} & \textcolor{orangered}{2} & \textcolor{orangered}{1} & \textcolor{orangered}{-1} \\ \textcolor{springgreen}{1} & \textcolor{springgreen}{2} & \textcolor{springgreen}{5} & \textcolor{springgreen}{2} & \textcolor{springgreen}{1} & \textcolor{springgreen}{-1} \\ 7 & 3 & 5 & 9 & 2 & 0 \\ 1 & 6 & 5 & -5 & 3 & 2 \\ \end{vmatrix} \\ \\ = & \begin{vmatrix} 1 & -2 & 5 & 0 & 0 & 0 \\ 3 & 8 & 1 & 0 & 0 & 0 \\ \textcolor{orangered}{4} & \textcolor{orangered}{-2} & \textcolor{orangered}{-8} & \textcolor{orangered}{0} & \textcolor{orangered}{0} & \textcolor{orangered}{0} \\ \textcolor{springgreen}{1} & \textcolor{springgreen}{2} & \textcolor{springgreen}{5} & \textcolor{springgreen}{2} & \textcolor{springgreen}{1} & \textcolor{springgreen}{-1} \\ 7 & 3 & 5 & 9 & 2 & 0 \\ 1 & 6 & 5 & -5 & 3 & 2 \\ \end{vmatrix} \\ \\ = & \begin{vmatrix} \textcolor{white}{\colorbox{green}{1}} & \textcolor{white}{\colorbox{green}{-2}} & \textcolor{white}{\colorbox{green}{5}} & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{0}} \\ \textcolor{white}{\colorbox{green}{3}} & \textcolor{white}{\colorbox{green}{8}} & \textcolor{white}{\colorbox{green}{1}} & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{0}} \\ \textcolor{white}{\colorbox{green}{4}} & \textcolor{white}{\colorbox{green}{-2}} & \textcolor{white}{\colorbox{green}{-8}} & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{0}} \\ 1 & 2 & 5 & \textcolor{white}{\colorbox{green}{2}} & \textcolor{white}{\colorbox{green}{1}} & \textcolor{white}{\colorbox{green}{-1}} \\ 7 & 3 & 5 & \textcolor{white}{\colorbox{green}{9}} & \textcolor{white}{\colorbox{green}{2}} & \textcolor{white}{\colorbox{green}{0}} \\ 1 & 6 & 5 & \textcolor{white}{\colorbox{green}{-5}} & \textcolor{white}{\colorbox{green}{3}} & \textcolor{white}{\colorbox{green}{2}} \\ \end{vmatrix} \end{aligned}

\begin{aligned} & \boldsymbol{K} = \\ \\ = & \begin{vmatrix} \textcolor{white}{\colorbox{green}{1}} & \textcolor{white}{\colorbox{green}{-2}} & \textcolor{white}{\colorbox{green}{5}} & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{0}} \\ \textcolor{white}{\colorbox{green}{3}} & \textcolor{white}{\colorbox{green}{8}} & \textcolor{white}{\colorbox{green}{1}} & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{0}} \\ \textcolor{white}{\colorbox{green}{4}} & \textcolor{white}{\colorbox{green}{-2}} & \textcolor{white}{\colorbox{green}{-8}} & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{0}} \\ 1 & 2 & 5 & \textcolor{white}{\colorbox{green}{2}} & \textcolor{white}{\colorbox{green}{1}} & \textcolor{white}{\colorbox{green}{-1}} \\ 7 & 3 & 5 & \textcolor{white}{\colorbox{green}{9}} & \textcolor{white}{\colorbox{green}{2}} & \textcolor{white}{\colorbox{green}{0}} \\ 1 & 6 & 5 & \textcolor{white}{\colorbox{green}{-5}} & \textcolor{white}{\colorbox{green}{3}} & \textcolor{white}{\colorbox{green}{2}} \\ \end{vmatrix} \\ \\ = & \begin{vmatrix} 1 & -2 & 5 \\ 3 & 8 & 1 \\ 4 & -2 & -8 \end{vmatrix} \cdot \begin{vmatrix} 2 & 1 & -1 \\ 9 & 2 & 0 \\ -5 & 3 & 2 \end{vmatrix} \\ \\ = & \begin{vmatrix} 0 & \textcolor{magenta}{-2} & 0 \\ 7 & 0 & 0 \\ 3 & 0 & \textcolor{magenta}{-22} \end{vmatrix} \cdot \begin{vmatrix} 0 & 1 & \textcolor{tan}{-1} \\ 5 & 2 & 0 \\ \textcolor{tan}{-11} & 0 & 5 \end{vmatrix} \\ \\ = & (\textcolor{magenta}{-2}) (\textcolor{magenta}{-22}) \cdot \begin{vmatrix} 0 & 1 & 0 \\ 7 & 0 & 0 \\ 3 & 0 & 1 \end{vmatrix} \cdot (\textcolor{tan}{-1}) (\textcolor{tan}{-11}) \begin{vmatrix} 0 & 1 & 1 \\ 5 & 2 & 0 \\ 1 & 0 & 5 \end{vmatrix} \\ \\ = & 44 \cdot (-7) \cdot 11 (-27) \\ \\ = & 44 \cdot 7 \cdot 11 \cdot 27 \\ \\ = & 484 \cdot 7\cdot 27 \\ \\ = & 91476 \end{aligned}