设 $\mathbf{A}=\begin{pmatrix}3&0\\1&1\end{pmatrix}$。

(a) 用数学归纳法证明，对于所有正整数 $n$，

(b) 求 $\mathbf{A}^n$ 的逆矩阵，用 $n$ 表示。

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。

先验n=1；归纳时将 $2A^k$ 右乘 $A$，左下元素合并后正好是下一次幂的目标值。

For $n=1$,

so the statement holds.

Assume it holds for some positive integer $k$, so

Then

This is the required expression for $n=k+1$. Since it holds for $n=1$ and truth for $n=k$ implies truth for $n=k+1$, the result follows by induction for all positive integers $n$.

先把上一问除以2得到A的n次幂，再用两行两列矩阵的逆公式。注意常数2在取逆前后的不同位置。

By (a),

Its determinant is

Therefore