IAL 2019 May FP1 Q5 | 國際數學站

Question

[!problem]

(a) Describe fully the single geometrical transformation represented by the matrix

$A = \begin{pmatrix} 1 & \sqrt{3} \\ \sqrt{3} & -1 \end{pmatrix} \times \frac{1}{2}$

(b) Hence write down the matrix $A^5$ .

(1)

The transformation represented by the matrix $C$ followed by the transformation represented by the matrix $B$ is equivalent to the transformation represented by the matrix $A$ .

$B = \begin{pmatrix} 2 & -7 \\ \sqrt{3} & 1 \end{pmatrix}$

(c) find the matrix $C$ , giving your answer in simplest form.

中文翻译

(a) 完整描述矩阵所表示的单一几何变换

$A = \begin{pmatrix} 1 & \sqrt{3} \\ \sqrt{3} & -1 \end{pmatrix} \times \frac{1}{2}$

(b) 由此写出矩阵 $A^5$ 。

(1)

矩阵 $C$ 所表示的变换后接矩阵 $B$ 所表示的变换等价于矩阵 $A$ 所表示的变换。

$B = \begin{pmatrix} 2 & -7 \\ \sqrt{3} & 1 \end{pmatrix}$

(c) 求矩阵 $C$ ，将答案写成最简形式。