https://file.aiquickdraw.com/mathbot/file/1730988721210ipj2glwl.jpg

Let's analyze each exercise one by one.

### Problem E7

We are given a set \( G \) of matrices:
\[
G = \left\{ A = \begin{pmatrix} a & b \\ b & a \end{pmatrix} \;|\; a, b \in \mathbb{R}, \det(A) \neq 0 \right\}.
\]
We need to prove that \( G \) forms a group under matrix multiplication. To do this, we must show that:
1. **Closure**: The product of any two matrices in \( G \) is also in \( G \).
2. **Associativity**: Matrix multiplication is associative.
3. **Identity Element**: There exists an identity element in \( G \).
4. **Inverse Element**: For each matrix in \( G \), there exists an inverse in \( G \).

### Problem E8

We have a set of matrices:
\[
\mathcal{M} = \left\{ \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}^n \;|\; n \in \mathbb{N}, n \geq 1 \right\} \subset M_3(\mathbb{C}).
\]

#### Part (a)
We need to prove that \( (\mathcal{M}, \cdot) \) is a commutative group, meaning that it satisfies the four group properties and is commutative.

#### Part (b)
We are asked to study whether the operation \( A \circ B = A^4 \cdot B^4 \), defined on \( \mathcal{M} \), gives it a group structure.

### Problem E9

We are given the set \( A_n = \{ \sigma \in S_n \;|\; \sigma \text{ is an even permutation} \} \).

#### Part (a)
We need to prove that \( (A_n, \circ) \) is a group. This means verifying the group properties with the set of even permutations under composition.

#### Part (b)
We need to determine for which values of \( n \) the group \( A_n \) is commutative.

Would you like a detailed solution for each part? Here are some further questions for exploration:

1. What are the specific elements in \( G \) that might fail the group properties?
2. How can we verify closure under matrix multiplication in \( G \)?
3. What are the implications of the operation \( A \circ B = A^4 \cdot B^4 \) for the set \( \mathcal{M} \)?
4. How do we determine if an even permutation set \( A_n \) is commutative for specific values of \( n \)?
5. What properties distinguish even and odd permutations in symmetric groups?

**Tip:** When proving group properties, carefully check closure and inverses as they are commonly tricky in matrix and permutation groups.

Demonstrate that certain sets of matrices and permutations form groups and explore their properties.

Explore group properties in sets of matrices and even permutations. This problem includes proving group closure, identity, associativity, and inverses, along with conditions for commutativity in matrix and permutation groups.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation