Does it have a Euler path?

Does it have a Hamilton circuit?

Does it have a Hamilton path?

2. (15%) Consider the following relations on the set of integers. Answer each of the three following questions and give a brief proof for each of your answers. $$R_1=\{(x, y) \mid (x - y)^2 < 10\}$$ $$R_2=\{(x, y) \mid x + y = 0\}$$ $$R_3=\{(x, y) \mid \gcd(x, y) = 1\}$$ $$R_4=\{(x, y) \mid (x - y) \bmod 5 = 0\}$$ $$R_5=\{(x, y) \mid x > y\}$$ Which of these relations are reflexive?

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. $$A = \begin{bmatrix} -2 & -6 & -6 \\ -3 & 2 & -2 \\ 3 & 2 & 6 \end{bmatrix}$$

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