对于这类不问“是什么”，而是问“不是什么”的题目要格外注意

一、题目

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) $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+4 x_{1} x_{2}$

(D) $2 x_{1}^{2}+2 x_{2}^{2}-x_{3}^{2}+2 x_{1} x_{2}$

难度评级：

(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。