一、题目![题目 - 荒原之梦](https://documents.zhaokaifeng.com/uploads/2017/06/06/f68a9e590526998388b0f9b71bd5d3f73dda4ed9764819fe8f36488fa537e9b9499f465fd201d7c117b8901c3ad071915a34a688058a739ebc39835753a8d7cc.svg)
$$
\begin{aligned}
& |\boldsymbol{K}| = \\ \\
& \begin{vmatrix}
1 & -2 & 5 & 0 & 0 & 0 \\
3 & 8 & 1 & 0 & 0 & 0 \\
5 & 0 & -3 & 2 & 1 & -1 \\
1 & 2 & 5 & 2 & 1 & -1 \\
7 & 3 & 5 & 9 & 2 & 0 \\
1 & 6 & 5 & -5 & 3 & 2 \\
\end{vmatrix} \\ \\
& = ?
\end{aligned}
$$
难度评级:
二、解析 ![解析 - 荒原之梦](https://documents.zhaokaifeng.com/uploads/2017/06/06/6fff698aa5c66c6c7a143e3d2a00fa8ee7eab76be5360d89eb43a03143848e8cd60377c76bf830c93ec6603be5af661d9c52238834792ea548bf14de10b05ad9.svg)
观察可知,行列式 $\boldsymbol{K}$ 的零元素主要集中在行列式的右上角:
$$
\begin{vmatrix}
1 & -2 & 5 & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{0}} \\
3 & 8 & 1 & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{0}} \\
5 & 0 & -3 & 2 & 1 & -1 \\
1 & 2 & 5 & 2 & 1 & -1 \\
7 & 3 & 5 & 9 & 2 & 0 \\
1 & 6 & 5 & -5 & 3 & 2 \\
\end{vmatrix}
$$
但是,该行列式的第 $3$ 行和第 $4$ 行存在一些相等的元素,刚好可以将第 $3$ 行的一些元素消为零:
$$
\begin{vmatrix}
1 & -2 & 5 & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{0}} \\
3 & 8 & 1 & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{0}} \\
5 & 0 & -3 & \textcolor{black}{\colorbox{tan}{2}} & \textcolor{black}{\colorbox{tan}{1}} & \textcolor{black}{\colorbox{tan}{-1}} \\
1 & 2 & 5 & \textcolor{black}{\colorbox{tan}{2}} & \textcolor{black}{\colorbox{tan}{1}} & \textcolor{black}{\colorbox{tan}{-1}} \\
7 & 3 & 5 & 9 & 2 & 0 \\
1 & 6 & 5 & -5 & 3 & 2 \\
\end{vmatrix}
$$
于是,在行列式 $\boldsymbol{K}$ 中,用第 $3$ 行减去第 $4$ 行,可得:
$$
\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}
$$
于是,根据行列中的“方块乘法”可知:
$$
\begin{aligned}
& \boldsymbol{K} = \\ \\
= & \begin{vmatrix}
\textcolor{white}{\colorbox{green}{1}} & \textcolor{white}{\colorbox{green}{-2}} & \textcolor{white}{\colorbox{green}{5}} & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{0}} \\
\textcolor{white}{\colorbox{green}{3}} & \textcolor{white}{\colorbox{green}{8}} & \textcolor{white}{\colorbox{green}{1}} & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{0}} \\
\textcolor{white}{\colorbox{green}{4}} & \textcolor{white}{\colorbox{green}{-2}} & \textcolor{white}{\colorbox{green}{-8}} & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{0}} & \textcolor{black}{\colorbox{pink}{0}} \\
1 & 2 & 5 & \textcolor{white}{\colorbox{green}{2}} & \textcolor{white}{\colorbox{green}{1}} & \textcolor{white}{\colorbox{green}{-1}} \\
7 & 3 & 5 & \textcolor{white}{\colorbox{green}{9}} & \textcolor{white}{\colorbox{green}{2}} & \textcolor{white}{\colorbox{green}{0}} \\
1 & 6 & 5 & \textcolor{white}{\colorbox{green}{-5}} & \textcolor{white}{\colorbox{green}{3}} & \textcolor{white}{\colorbox{green}{2}} \\
\end{vmatrix} \\ \\
= & \begin{vmatrix}
1 & -2 & 5 \\
3 & 8 & 1 \\
4 & -2 & -8
\end{vmatrix} \cdot \begin{vmatrix}
2 & 1 & -1 \\
9 & 2 & 0 \\
-5 & 3 & 2
\end{vmatrix} \\ \\
= & \begin{vmatrix}
0 & \textcolor{magenta}{-2} & 0 \\
7 & 0 & 0 \\
3 & 0 & \textcolor{magenta}{-22}
\end{vmatrix} \cdot \begin{vmatrix}
0 & 1 & \textcolor{tan}{-1} \\
5 & 2 & 0 \\
\textcolor{tan}{-11} & 0 & 5
\end{vmatrix} \\ \\
= & (\textcolor{magenta}{-2}) (\textcolor{magenta}{-22}) \cdot \begin{vmatrix}
0 & 1 & 0 \\
7 & 0 & 0 \\
3 & 0 & 1
\end{vmatrix} \cdot (\textcolor{tan}{-1}) (\textcolor{tan}{-11}) \begin{vmatrix}
0 & 1 & 1 \\
5 & 2 & 0 \\
1 & 0 & 5
\end{vmatrix} \\ \\
= & 44 \cdot (-7) \cdot 11 (-27) \\ \\
= & 44 \cdot 7 \cdot 11 \cdot 27 \\ \\
= & 484 \cdot 7\cdot 27 \\ \\
= & 91476
\end{aligned}
$$
高等数学![箭头 - 荒原之梦](https://documents.zhaokaifeng.com/uploads/2017/06/06/c19692009799eac2a7eb5b9d73167ae3dd6cad169ea3ccdbeb97491b80e87593cfa7384844ec1720d0fb9cf5f00ac456f249d047b61ce2d90bdd241e042f4d89.svg)
涵盖高等数学基础概念、解题技巧等内容,图文并茂,计算过程清晰严谨。
线性代数![箭头 - 荒原之梦](https://documents.zhaokaifeng.com/uploads/2017/06/06/c19692009799eac2a7eb5b9d73167ae3dd6cad169ea3ccdbeb97491b80e87593cfa7384844ec1720d0fb9cf5f00ac456f249d047b61ce2d90bdd241e042f4d89.svg)
以独特的视角解析线性代数,让繁复的知识变得直观明了。
特别专题![箭头 - 荒原之梦](https://documents.zhaokaifeng.com/uploads/2017/06/06/c19692009799eac2a7eb5b9d73167ae3dd6cad169ea3ccdbeb97491b80e87593cfa7384844ec1720d0fb9cf5f00ac456f249d047b61ce2d90bdd241e042f4d89.svg)
通过专题的形式对数学知识结构做必要的补充,使所学知识更加连贯坚实。