Math Problem Statement
. Consider the set G of 3X3 matrices defined as follows:
10
10
C01
2
We define, as the G..) an abelian group? G= [[1, x, x], [0, 1, y], [0, 0, 1]] |x,y,x in mathbb R \
Solution
We are tasked with analyzing the set of matrices of the form:
1 & x & x \\ 0 & 1 & y \\ 0 & 0 & 1 \end{bmatrix} \; \big| \; x, y \in \mathbb{R} \right\}.$$ and determining whether $$G$$ forms an **abelian group** under matrix multiplication. --- ### **Step 1: Group Properties** To determine if $$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$$. --- ### **Step 2: Closure** Take two arbitrary matrices $$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{Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Group Theory
Matrix Algebra
Abelian Groups
Matrix Multiplication
Formulas
Matrix multiplication rules
Definition of an abelian group
Theorems
Group properties
Commutativity of abelian groups
Suitable Grade Level
Undergraduate Mathematics