一、题目![题目 - 荒原之梦](https://documents.zhaokaifeng.com/uploads/2017/06/06/f68a9e590526998388b0f9b71bd5d3f73dda4ed9764819fe8f36488fa537e9b9499f465fd201d7c117b8901c3ad071915a34a688058a739ebc39835753a8d7cc.svg)
下面的二次型中,经正交变换后得到的标准形不是 $y_{1}^{2}+3 y_{2}^{2}-y_{3}^{2}$ 的是哪个?
(A) $3 x_{2}^{2}+2 x_{1} x_{3}$
(B) $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+4 x_{1} x_{2}$
(C) $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4 x_{1} x_{2}-4 x_{2} x_{3}$
(D) $2 x_{1}^{2}+2 x_{2}^{2}-x_{3}^{2}+2 x_{1} x_{2}$
难度评级:
二、解析 ![解析 - 荒原之梦](https://documents.zhaokaifeng.com/uploads/2017/06/06/6fff698aa5c66c6c7a143e3d2a00fa8ee7eab76be5360d89eb43a03143848e8cd60377c76bf830c93ec6603be5af661d9c52238834792ea548bf14de10b05ad9.svg)
(A):
$$
\left[\begin{array}{lll}
0 & 0 & 1 \\
0 & 3 & 0 \\
1 & 0 & 0
\end{array}\right] \Rightarrow|\lambda E – A|=0 \Rightarrow
$$
$$
\left|\begin{array}{ccc}
\lambda & 0 & -1 \\
0 & \lambda-3 & 0 \\
-1 & 0 & \lambda
\end{array}\right|=0
\Rightarrow
$$
$$
\lambda^{2}(\lambda-3)-(\lambda-3)=0 \Rightarrow
$$
$$
(\lambda-3)(\lambda+1)(\lambda-1)=0 \Rightarrow
$$
$$
\lambda=3, \ -1, \ 1
$$
(B):
$$
\left[\begin{array}{lll}
1 & 2 & 0 \\
2 & 1 & 0 \\
0 & 0 & 1
\end{array}\right] \Rightarrow
$$
$$
\left|\begin{array}{ccc}
\lambda-1 & -2 & 0 \\
-2 & \lambda-1 & 0 \\
0 & 0 & \lambda-1
\end{array}\right|=0 \Rightarrow
$$
$$
(\lambda-1)^{3}-4(\lambda-1)=0 \Rightarrow
$$
$$
(\lambda-1)\left[\lambda^{2}-2 \lambda-3\right] \Rightarrow
$$
$$
(\lambda-1)(\lambda-3)(r+1)=0 \Rightarrow
$$
$$
\lambda=1, \ 3, \ -1
$$
(C):
$$
\left[\begin{array}{ccc}
1 & -2 & 0 \\
-2 & 1 & -2 \\
0 & -2 & 1
\end{array}\right] \Rightarrow
$$
$$
\left|\begin{array}{ccc}
\lambda-1 & 2 & 0 \\
2 & \lambda-1 & 2 \\
0 & 2 & \lambda-1
\end{array}\right|=0 \Rightarrow
$$
$$
(\lambda-1)^{3}-4(\lambda-1)-4(\lambda-1)=0 \Rightarrow
$$
$$
(\lambda-1)\left[(\lambda-1)^{2}-8\right]=0 \Rightarrow
$$
$$
(\lambda-1)\left(\lambda^{2}-2 \lambda-7\right) \neq
(\lambda-1)(\lambda+1)(\lambda-3)
$$
(D):
$$
\left[\begin{array}{ccc}
2 & 1 & 0 \\
1 & 2 & 0 \\
0 & 0 & -1
\end{array}\right] \Rightarrow
$$
$$
\left|\begin{array}{ccc}
\lambda-2 & -1 & 0 \\
-1 & \lambda-2 & 0 \\
0 & 0 & \lambda+1
\end{array}\right|=0 \Rightarrow
$$
$$
(\lambda+1)(\lambda-2)^{2}-(\lambda+1)=0 \Rightarrow
$$
$$
(\lambda+1)\left(\lambda^{2}-4 \lambda+3\right)=0 \Rightarrow
$$
$$
(\lambda+1)(\lambda-3)(\lambda-1)=0 \Rightarrow
$$
$$
\lambda = -1, \ 3, \ 1
$$
综上可知,(C) 选项符合题意。
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。