一、题目![题目 - 荒原之梦](https://documents.zhaokaifeng.com/uploads/2017/06/06/f68a9e590526998388b0f9b71bd5d3f73dda4ed9764819fe8f36488fa537e9b9499f465fd201d7c117b8901c3ad071915a34a688058a739ebc39835753a8d7cc.svg)
已知 $A$ 是 $n$ 阶正交矩阵,若 $|A|$ $=$ $1$, 请证明当 $n$ 为奇数时,有 $| \boldsymbol { E } – \boldsymbol { A } |$ $=$ $0$.
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二、解析 ![解析 - 荒原之梦](https://documents.zhaokaifeng.com/uploads/2017/06/06/6fff698aa5c66c6c7a143e3d2a00fa8ee7eab76be5360d89eb43a03143848e8cd60377c76bf830c93ec6603be5af661d9c52238834792ea548bf14de10b05ad9.svg)
由于 $A$ 是正交矩阵,所以:
$$
A A ^{\top} = A ^{\top} A = E
$$
又因为 $\textcolor{orangered}{|A|}$ $\textcolor{orangered}{=}$ $\textcolor{orangered}{1}$, 而且矩阵乘法越多越有利于解题,所以,可以利用转置运算引入矩阵乘法:
$$
\begin{aligned}
\textcolor{springgreen}{| \boldsymbol { E } – \boldsymbol { A } |} \\ \\
& = \left| \boldsymbol { A } \boldsymbol { A } ^ { \top } – \boldsymbol { A } \right| \\ \\
& = \left| \boldsymbol { A } \left( \boldsymbol { A } ^ { \top } – \boldsymbol { E } \right) \right| \\ \\
& = \textcolor{orangered}{| \boldsymbol { A } |} \cdot \left| \boldsymbol { A } ^ { \top } – \boldsymbol { E } \right| \\ \\
& = \textcolor{orangered}{1} \cdot \left| \boldsymbol { A } ^ { \top } – \boldsymbol { E } \right| \\ \\
& = \textcolor{pink}{\left| \boldsymbol { A } ^ { \top } – \boldsymbol { E } \right| }
\end{aligned}
$$
又因为 $E ^{\top}$ $=$ $E$, 且转置运算不会改变行列式的值,所以:
$$
\begin{aligned}
\textcolor{springgreen}{| \boldsymbol { E } – \boldsymbol { A } |} = \textcolor{pink}{\left| \boldsymbol { A } ^ { \top } – \boldsymbol { E } \right| } \\ \\
& = | ( \boldsymbol { A } – \boldsymbol { E } ) ^ { \top } | \\ \\
& = \textcolor{yellow}{| \boldsymbol { A } – \boldsymbol { E } |}
\end{aligned}
$$
由于 $n$ 是奇数,又根据「荒原之梦考研数学网」的《行列式 $|A-B|$ 和 $|B-A|$ 有什么关系》这篇文章可知:
$$
\begin{aligned}
\textcolor{springgreen}{| \boldsymbol { E } – \boldsymbol { A } |} = \textcolor{yellow}{| \boldsymbol { A } – \boldsymbol { E } |} \\ \\
& = ( – 1 ) ^ { n } | \boldsymbol { E } – \boldsymbol { A } | \\ \\
& = \textcolor{tan}{ – | \boldsymbol { E } – \boldsymbol { A } | }
\end{aligned}
$$
即:
$$
|E – A| = – |E-A|
$$
所以:
$$
\textcolor{green}{\boldsymbol{
| \boldsymbol { E } – \boldsymbol { A } | = 0
}}
$$
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。