. Consider the set G of 3X3 matrices defined as follows:

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C01

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We define, as the G..) an abelian group? G= [[1, x, x], [0, 1, y], [0, 0, 1]] |x,y,x in mathbb R \

We are tasked with analyzing the set \( G \) of \( 3 \times 3 \) matrices of the form:

\[
G = \left\{ \begin{bmatrix} 
1 & x & x \\ 
0 & 1 & y \\ 
0 & 0 & 1 
\end{bmatrix} \; \big| \; x, y \in \mathbb{R} \right\}.
\]

and determining whether \( G \) forms an **abelian group** under matrix multiplication.

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### **Step 1: Group Properties**
To determine if \( G \) forms an abelian group under matrix multiplication, we must verify the following properties:

1. **Closure**: If \( A, B \in G \), then \( AB \in G \).
2. **Associativity**: Matrix multiplication is associative (always true for matrices).
3. **Identity**: There exists an identity matrix \( I \in G \) such that \( IA = AI = A \) for all \( A \in G \).
4. **Inverse**: For every \( A \in G \), there exists an inverse \( A^{-1} \in G \).
5. **Commutativity**: \( AB = BA \) for all \( A, B \in G \).

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### **Step 2: Closure**

Take two arbitrary matrices \( A, B \in G \) where:

\[
A = \begin{bmatrix} 
1 & x_1 & x_1 \\ 
0 & 1 & y_1 \\ 
0 & 0 & 1 
\end{bmatrix}, \quad
B = \begin{bmatrix} 
1 & x_2 & x_2 \\ 
0 & 1 & y_2 \\ 
0 & 0 & 1 
\end{bmatrix}.
\]

Compute \( AB \):
\[
AB = \begin{bmatrix} 
1 & x_1 & x_1 \\ 
0 & 1 & y_1 \\ 
0 & 0 & 1 
\end{bmatrix}
\begin{bmatrix} 
1 & x_2 & x_2 \\ 
0 & 1 & y_2 \\ 
0 & 0 & 1 
\end{

Consider the set G of 3X3 matrices defined as follows: G = [[1, x, x], [0, 1, y], [0, 0, 1]] |x,y,x in mathbb R. We need to determine whether G forms an abelian group.

This page explores the set G of 3x3 matrices of the form [[1, x, x], [0, 1, y], [0, 0, 1]] and determines whether it forms an abelian group under matrix multiplication. The solution includes an analysis of group properties such as closure, associativity, identity, inverse, and commutativity. Ideal for students studying group theory and matrix algebra.

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