矩阵 $\mathbf A$ 为

(a) 求使 $\mathbf A$ 为非奇异矩阵的 $k$ 的取值集合。

(b) 已知 $\mathbf A$ 为非奇异矩阵，用 $k$ 表示 $\mathbf A^{-1}$ 第一行的各元素。

(c) 已知 $\mathbf B=\begin{pmatrix}1&0&0\\0&1&0\end{pmatrix}$，给出一个矩阵 $\mathbf C$ 的例子，使得 $\mathbf B\mathbf A\mathbf C=\begin{pmatrix}2&1\\k&4\end{pmatrix}$。

(d) 求使矩阵 $\begin{pmatrix}2&1\\k&4\end{pmatrix}$ 所表示的 $x$–$y$ 平面内的变换具有两条经过原点的不同不变直线的 $k$ 的取值集合。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

按第一行展开行列式，并把“非奇异”转成行列式不为零。

Expanding along the first row,

Thus $\mathbf A$ is non-singular when $60-12k\ne0$, giving $\boxed{k\ne5}$.

逆矩阵第一行来自余因子矩阵的第一列；注意转置与第二个余因子的负号。

Since $\mathbf A^{-1}=(\det\mathbf A)^{-1}(\operatorname{cof}\mathbf A)^{\mathsf T}$, its top row uses the cofactors $C_{11},C_{21},C_{31}$:

Hence the required row is

左乘 $B$ 选出前两行；右边只需把所得矩阵的前两列交换，并舍去第三列。直接写出实现这个操作的矩阵并验算。

First,

Choose

Then

as required.

先求不变方向的两个实斜率，再排除其中一条直线被压成原点的情形；两个不同的实斜率本身还不够。

A nonvertical line through the origin has equation $y=mx$. Under the transformation,

For its image to be the same line, $k+4m=m(2+m)$, with $2+m\ne0$. Thus

Two distinct real slopes require

Their values are $m=1\pm\sqrt{1+k}$, and the scale factors along these directions are

For $k>-1$, one of these vanishes precisely when $k=8$. At $k=8$, the direction $m=-2$ is mapped entirely to the origin, so it does not give a line mapped onto itself. This value must be excluded.

The vertical line is not invariant: $(0,y)$ maps to $(y,4y)$, which is not vertical. Therefore no additional invariant line was omitted, and the required set is