Which of these relations are symmetric?

Which of these relations are transitive?

3. (10%) A derangement is a permutation of objects that leaves no object in the original position. For example, the permutation 21453 is a derangement of 12345 because no digit is left in its original position. What is the total number of derangements of 1234? List all derangements of 1234.

4. (10%) Find all integer solutions for $x$ such that $18x \equiv 6 \pmod{141}$.

5. (20%) In the following, you should use uppercase letters to denote matrices and lowercase letters with subscripts to denote their entries. For example, if M is a matrix, then $m_{ij}$ denotes its (i,j)-entry. Use lowercase letters with an arrow to denote column vectors of a matrix. For example, $\vec{m}_i$ denotes the i-th column vector of the matrix M. In your proofs, use the following definitions as needed. Definition (Equality of matrices): Matrices A and B are equal if they have the same size and have equal corresponding entries $a_{ij} = b_{ij}$ for all (i,j). Definition (Matrix transpose): The transpose of an $m \times n$ matrix A is the $n \times m$ matrix denoted by $A^T$ whose (i,j)-entry is the (j,i)-entry of A. Definition (Linear combination): A linear combination of vectors $\vec{u}_1, \vec{u}_2, ..., \vec{u}_k$ is a vector of the form $c_1\vec{u}_1 + c_2\vec{u}_2 + \cdots + c_k\vec{u}_k$, where $c_1, c_2, ..., c_k$ are scalars. Definition (Standard vectors): $$\vec{e}_1 = \begin{bmatrix} 1 \\ 0 \\ \vdots \\ 0 \end{bmatrix}, \vec{e}_2 = \begin{bmatrix} 0 \\ 1 \\ \vdots \\ 0 \end{bmatrix}, ..., \vec{e}_n = \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 1 \end{bmatrix}$$ Now let A denote a $m \times n$ matrix. Prove that $(A^T)^T = A$ and $A\vec{e}_i = \vec{a}_i$ for $i = 1, ..., n.

6. (10%) Use Gauss elimination to find the inverse of the following matrix A. First, reduce the matrix to row echelon form, and then continue to reduced row echelon form to obtain the inverse matrix. $$A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 5 & 6 \\ 3 & 4 & 8 \end{bmatrix}$$

7. (20%) Find the eigenvalues of the following matrix A and determine a basis for each eigenspace. Use the characteristic polynomials and Gauss elimination to obtain your answers. $$A = \begin{bmatrix} -2 & -6 & -6 \\ -3 & 2 & -2 \\ 3 & 2 & 6 \end{bmatrix}$$

1.(1) 台電公司向甲廠商購買 A公司生產之X型號貨品，訂約後始發現該貨品早已停產，請問甲廠商不依約交貨有無理由?(3分)

(2) 承(1)所述，倘甲廠商明知該貨品已停產而仍與台電公司訂約，則甲廠商無法依約交貨致台電公司受有損害，台電公司得否請求損害賠償?(3分)

(3) 台電公司招標時，非因本身之過失，將性質不同之A貨品誤為欲採購之B貨品，乙廠商明知此事，仍投標而得標，請問台電公司有無救濟之方式及須否承擔損害賠償責任?(12分)

(4) 承(3)所述，倘乙廠商係因過失而不知台電公司誤認之事，則法院得否減輕或免除台電公司之賠償金額?(3分)

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