110年 - 110 國立臺北大學_碩士班一般入學考試試題_通訊工程學系：工程數學(線性代數與機率)#147263

1. (10%) 請使用 Gauss-Jordan Elimination 來求出(1)中 $A$ 的 inverse matrix，也就是求 $A^{-1}$。請將計算過程寫出，沒有計算過程不計分。 $$A = \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 2 & 1 \\ 1 & 2 & 3 & 4 \\ 3 & 3 & 1 & 1 \end{bmatrix} \tag{1}$$

2. (10%) $$\text{已知 } A = \begin{bmatrix} 2 & 3 \\ 4 & 1 \end{bmatrix} = \begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{-3}{5} \\ \frac{1}{\sqrt{2}} & \frac{4}{5} \end{bmatrix} \begin{bmatrix} 5 & 0 \\ 0 & -2 \end{bmatrix} \begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{-3}{5} \\ \frac{1}{\sqrt{2}} & \frac{4}{5} \end{bmatrix}^{-1} \tag{2}$$ 請利用(2)的結果來求取 $\begin{bmatrix} 0.4 & 0.6 \\ 0.8 & 0.2 \end{bmatrix}^4$

3. (10%) For which number $a$ will make the matrix $A$ singular? $A = \begin{bmatrix} a & 3 & 4 \\ a & 1 & -2 \\ a & a & a \end{bmatrix}$

4. (10%)(5% for Gram-Schmidt Process and 5% for matrix arrangement) 請使用 QR decomposition 將以下的 A 扮解成一個包含 orthonormal column vectors 的 $Q$ 以及一個 upper triangular matrix $R$，也就是 $A = QR$，其中 $Q^T Q = I \cdot A(:,1) = cQ(:,1)$ 且 $c$ 為一個常數。請將計算過程寫出，沒有計算過程不計分。 $$A = \begin{bmatrix} 1 & 1 & -1 & 1 \\ 2 & -3 & 3 & -3 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 3 & -4 \end{bmatrix}$$ Hints : (a)從 $A(:,1)$ 開始做 Gram-Schmidt Process，也就是 $A(:,1) = cQ(:,1)$，其中 $c$ 為一個常數。

5. (10%) $$\text{Let } A = \begin{bmatrix} 0 & 0 & 3 & -4 \\ 2 & -3 & 3 & -3 \\ 0 & 0 & 1 & 2 \\ 1 & 1 & -1 & 1 \end{bmatrix}^T \text{ and } B = \begin{bmatrix} 1 & 2 & -1 & 2 \\ -3 & 4 & 3 & -6 \\ 0 & 0 & 1 & 4 \\ 0 & 0 & 3 & -8 \end{bmatrix} \text{.please find |AB|}.$$

(a) What is the PDF of the random variable \( X \)?

(b) What is the expectation of the random variable \( X \)?

(c) What is the CDF of \( \min(X, Y) \)?

(a) What is the constant \( c \)?

(b) What is the probability \( P[2 \leq X \leq 8] \)?

(c) What is the expectation of \( X^2 \)?

(a) Consider two events \( E_{1} \) and \( E_{2} \). Is the statement "if \( E_{1} \) and \( E_{2} \) are disjoint, then \( E_{1} \) and \( E_{2} \) are independent" true?

(b) Consider two events \( E_{1} \) and \( E_{2} \). Is the statement " \( P[E_{1} \cup E_{2}] = P[E_{1}] + P[E_{2}] \)" true?

(c) Consider a continuous random variable \( X \). Is the statement " \( P[X = 0] = 0 \) " true?

(d) Is the function \( f(x) = \begin{cases} e^{-2x} & \text{if } x \geq 0; \\ 0 & \text{else}, \end{cases} \) a feasible PDF of a continuous random variable?