一、题目![题目 - 荒原之梦](https://documents.zhaokaifeng.com/uploads/2017/06/06/f68a9e590526998388b0f9b71bd5d3f73dda4ed9764819fe8f36488fa537e9b9499f465fd201d7c117b8901c3ad071915a34a688058a739ebc39835753a8d7cc.svg)
设 $\boldsymbol{A}$ 为三阶矩阵, $\boldsymbol{P}=\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 1\end{array}\right)$, 若 $\boldsymbol{P}^{\mathrm{\top}} \boldsymbol{A} \boldsymbol{P}^{2}=\left(\begin{array}{ccc}a+2 c & 0 & c \\ 0 & b & 0 \\ 2 c & 0 & c\end{array}\right)$, 则 $\boldsymbol{A}=$
A. $\left(\begin{array}{lll}c & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & b\end{array}\right)$
B. $\left(\begin{array}{lll}b & 0 & 0 \\ 0 & c & 0 \\ 0 & 0 & a\end{array}\right)$
C. $\left(\begin{array}{lll}a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c\end{array}\right)$
D. $\left(\begin{array}{lll}c & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & a\end{array}\right)$
难度评级:
二、解析 ![解析 - 荒原之梦](https://documents.zhaokaifeng.com/uploads/2017/06/06/6fff698aa5c66c6c7a143e3d2a00fa8ee7eab76be5360d89eb43a03143848e8cd60377c76bf830c93ec6603be5af661d9c52238834792ea548bf14de10b05ad9.svg)
由于:
$$
\boldsymbol{A} = (\boldsymbol{P}^{\top})^{-1} \quad (\boldsymbol{P}^{\top} \boldsymbol{A} \boldsymbol{P}^{2}) \quad (\boldsymbol{P}^{2})^{-1}
$$
且,由求逆变换法,可得:
$$
\boldsymbol{P}^{\mathrm{\top}}=\left(\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right) \Rightarrow \left(\boldsymbol{P}^{\mathrm{\top}}\right)^{-1}=\left(\begin{array}{lll}1 & 0 & -1 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right)
$$
$$
\boldsymbol{P}^{2}=\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 2 & 0 & 1\end{array}\right) \Rightarrow \left(\boldsymbol{P}^{2}\right)^{-1}=\left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ -2 & 0 & 1\end{array}\right)
$$
于是:
$$
\begin{aligned} \boldsymbol{A} & =\left(\boldsymbol{P}^{\mathrm{\top}}\right)^{-1}\left(\begin{array}{ccc}a+2 c & 0 & c \\ 0 & b & 0 \\ 2 c & 0 & c\end{array}\right)\left(\boldsymbol{P}^{2}\right)^{-1} \\ \\
& =\left(\begin{array}{ccc}1 & 0 & -1 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right)\left(\begin{array}{ccc}a+2 c & 0 & c \\ 0 & b & 0 \\ 2 c & 0 & c\end{array}\right)\left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ -2 & 0 & 1\end{array}\right) \\ \\
& =\left(\begin{array}{lll}a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c\end{array}\right) \end{aligned}
$$
综上可知,本题应选 C .
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。