8. Let $$ A=\left[\begin{array}{rrr}4 & -1 & 6 \ 2 & 1 & 6 \ 2 & -1 & 8\end{array} ight] $$. An eigenvalue of $$ A $$ is 2 . Find a basis for the corresponding eigenspace. 9. Show that set of integers $$ Z $$ is a abelian group under addition.

Question

8. Let $$ A=\left[\begin{array}{rrr}4 & -1 & 6 \ 2 & 1 & 6 \ 2 & -1 & 8\end{array}ight] $$. An eigenvalue of $$ A $$ is 2 . Find a basis for the corresponding eigenspace. 9. Show that set of integers $$ Z $$ is a abelian group under addition.

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To solve the problems, we will address each one separately.

### Problem 8: Finding the Eigenspace Corresponding to Eigenvalue 2

Given the matrix

$$
A = \begin{bmatrix} 4 & -1 & 6 \ 2 & 1 & 6 \ 2 & -1 & 8 \end{bmatrix}
$$

and the eigenvalue $$ \lambda = 2 $$, we need to find the eigenspace corresponding to this eigenvalue. The eigenspace is found by solving the equation

$$
(A - \lambda I) \mathbf{v} = 0
$$

where $$ I $$ is the identity matrix. First, we compute $$ A - 2I $$:

$$
A - 2I = \begin{bmatrix} 4 & -1 & 6 \ 2 & 1 & 6 \ 2 & -1 & 8 \end{bmatrix} - \begin{bmatrix} 2 & 0 & 0 \ 0 & 2 & 0 \ 0 & 0 & 2 \end{bmatrix} = \begin{bmatrix} 2 & -1 & 6 \ 2 & -1 & 6 \ 2 & -1 & 6 \end{bmatrix}
$$

Next, we set up the equation

$$
\begin{bmatrix} 2 & -1 & 6 \ 2 & -1 & 6 \ 2 & -1 & 6 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix} = \begin{bmatrix} 0 \ 0 \ 0 \end{bmatrix}
$$

This leads to the system of equations:

1. $$ 2x_1 - x_2 + 6x_3 = 0 $$

Since the rows of the matrix are linearly dependent, we can reduce this to a single equation. We can express $$ x_2 $$ in terms of $$ x_1 $$ and $$ x_3 $$:

$$
x_2 = 2x_1 + 6x_3
$$

Now, we can express the general solution in terms of free variables. Let $$ x_3 = t $$ (a free variable) and $$ x_1 = s $$ (another free variable). Then we have:

$$
x_2 = 2s + 6t
$$

Thus, the general solution can be written as:

$$
\begin{bmatrix} x_1 \ x_2 \ x_3 \end{bmatrix} = \begin{bmatrix} s \ 2s + 6t \ t \end{bmatrix} = s \begin{bmatrix} 1 \ 2 \ 0 \end{bmatrix} + t \begin{bmatrix} 0 \ 6 \ 1 \end{bmatrix}
$$

The vectors $$ \begin{bmatrix} 1 \ 2 \ 0 \end{bmatrix} $$ and $$ \begin{bmatrix} 0 \ 6 \ 1 \end{bmatrix} $$ form a basis for the eigenspace corresponding to the eigenvalue $$ \lambda = 2 $$.

### Conclusion for Problem 8:
A basis for the eigenspace corresponding to the eigenvalue $$ 2 $$ is

$$
\left\{ \begin{bmatrix} 1 \ 2 \ 0 \end{bmatrix}, \begin{bmatrix} 0 \ 6 \ 1 \end{bmatrix} \right\}
$$

---

### Problem 9: Showing that the Set of Integers $$ \mathbb{Z} $$ is an Abelian Group Under Addition

To show that $$ \mathbb{Z} $$ is an abelian group under addition, we need to verify the following group properties:

1. **Closure**: For any $$ a, b \in \mathbb{Z} $$, the sum $$ a + b $$ is also in $$ \mathbb{Z} $$.
2. **Associativity**: For any $$ a, b, c \in \mathbb{Z} $$, $$ (a + b) + c = a + (b + c) $$.
3. **Identity Element**: There exists an element $$ e \in \mathbb{Z} $$ such that for any $$ a \in \mathbb{Z} $$, $$ a + e = a $$. The identity element is $$ 0 $$.
4. **Inverse Element**: For every $$ a \in \mathbb{Z} $$, there exists an element $$ b \in \mathbb{Z} $$ such that $$ a + b = 0 $$. The inverse of $$ a $$ is $$ -a $$.
5. **Commutativity**: For any $$ a, b \in \mathbb{Z} $$, $$ a + b = b + a $$.

Now, we verify each property:

1. **Closure**: If $$ a, b \in \mathbb{Z} $$, then $$ a + b $$ is an integer, so $$ \mathbb{Z} $$ is closed under addition.
2. **Associativity**: Addition of integers is associative: $$ (a + b) + c = a + (b + c) $$.
3. **Identity Element**: The integer $$ 0 $$ acts as the identity since $$ a + 0 = a $$ for any integer $$ a $$.
4. **Inverse Element**: For any integer $$ a $$, the integer $$ -a $$ is its inverse since $$ a + (-a) = 0 $$.
5. **Commutativity**: Addition of integers is commutative: $$ a + b = b + a $$.

### Conclusion for Problem 9:
Since all group properties are satisfied, $$ \mathbb{Z} $$ is an abelian group under addition.
For Problem 8, a basis for the eigenspace corresponding to the eigenvalue $$ 2 $$ is $$ \left\{ \begin{bmatrix} 1 \ 2 \ 0 \end{bmatrix}, \begin{bmatrix} 0 \ 6 \ 1 \end{bmatrix} \right\} $$. For Problem 9, $$ \mathbb{Z} $$ is an abelian group under addition.

Let . An eigenvalue of is 2 . Find a basis for the corresponding eigenspace.

Show that set of integers is a abelian group under addition.

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Let . An eigenvalue of is 2 . Find a basis for the corresponding eigenspace. Show that set of integers is a abelian group under addition.