Proving matrix group under addition for associative axiom

In summary, the conversation discussed a proof showing that a set of 2x2 square matrices with integral coefficients and all entries being the same is an abelian group under addition. The proof involved demonstrating the properties of associativity, identity, and inverse elements, and also mentioned an alternative proof using an isomorphism.
  • #1
cbarker1
Gold Member
MHB
347
23
Dear Everyone,

I have some feeling some uncertainty proving one of the axioms for a group. Here is the proof to show this is a group:
Let the set T be defined as a set of 2x2 square matrices with coefficients of integral values and all the entries are the same.
We want to show that T is an abelian group under addition. Let's denote that $(x)=\left[\begin{array}{cc}x&x\\x&x\end{array}\right]$.
Let $(a)$ and $(b)$ $\in T$ where (a),(b) are a arbitrary elements in the set T. Then by the definition of (a) and (b) as well the definition of matrix addition, So, \begin{equation}
(a)+(b)=\left[\begin{array}{cc}
a & a \\
a & a
\end{array}\right]+\left[\begin{array}{cc}
b & b \\
b & b
\end{array}\right]=\left[\begin{array}{cc}
a+b & a+b \\
a+b & a+b
\end{array}\right]= (a+b)\in T. \end{equation}
Let $(a)$ be an arbitrary element in the set $T$. Let $(e)=(0)$. Then, by the definition of the binary operation,
\begin{equation}
(a)+(0)=\left[\begin{array}{cc}
a & a \\
a & a
\end{array}\right]+\left[\begin{array}{cc}
0 & 0 \\
0 & 0
\end{array}\right]=\left[\begin{array}{cc}
a+0 & a+0 \\
a+0 & a+0\end{array}\right]=\left[\begin{array}{cc}
a & a \\
a & a
\end{array}\right]=(a).\end{equation} And \begin{equation}
(0)+(a)=\left[\begin{array}{cc}
0 & 0 \\
0 & 0
\end{array}\right]+\left[\begin{array}{cc}
a & a \\
a & a
\end{array}\right]=\left[\begin{array}{cc}
0+a & 0+a \\
0+a & 0+a\end{array}\right]=\left[\begin{array}{cc}
a & a \\
a & a
\end{array}\right]=(a)
\end{equation}
Let $(a)$ be an arbitrary element in the set $T$. Let $(a')=(-a)$. Since the coefficients of the matrix is the set of integers, so the additive inverse for the integers is the negative element of the set of integers. Then, by the definition of the binary operation, \begin{equation}
(a)+(-a)=\left[\begin{array}{cc}
a & a \\
a & a
\end{array}\right]+\left[\begin{array}{cc}
-a & -a \\
-a & -a
\end{array}\right]=\left[\begin{array}{cc}
a-a & a-a \\
a-a & a-a\end{array}\right]=\left[\begin{array}{cc}
0 & 0 \\
0 & 0
\end{array}\right]=(0).
\end{equation}
And \begin{equation}
(-a)+(a)=\left[\begin{array}{cc}
-a & -a \\
-a & -a
\end{array}\right]+\left[\begin{array}{cc}
a & a \\
a & a
\end{array}\right]=\left[\begin{array}{cc}
-a+a & -a+a \\
-a+a & -a+a\end{array}\right]=\left[\begin{array}{cc}
0 & 0 \\
0 & 0
\end{array}\right]=(0).
\end{equation}
Let $(a)$,$(b)$,$(c)$ $\in T$ where $(a),(b),(c)$ are a arbitrary elements in the set T. Then by the definition of (a),(b),(b) as well the definition of matrix addition, So, \begin{equation}
((a)+(b))+(c)=(\left[\begin{array}{cc}
a & a \\
a & a
\end{array}\right]+\left[\begin{array}{cc}
b & b \\
b & b
\end{array}\right])+\left[\begin{array}{cc}
c & c \\
c & c
\end{array}\right]=\left[\begin{array}{cc}
a+b & a+b \\
a+b & a+b
\end{array}\right]+\left[\begin{array}{cc}
c & c \\
c & c
\end{array}\right]=\left[\begin{array}{cc}
(a+b)+c & (a+b)+c \\
(a+b)+c & (a+b)+c
\end{array}\right]=\left[\begin{array}{cc}
(a+0+b)+c & (a+0+b)+c \\
(a+0+b)+c & (a+0+b)+c
\end{array}\right] =\left[\begin{array}{cc}
(a+0)+(b+c) & (a+0)+(b+c) \\
(a+0)+(b+c )& (a+0)+(b+c)
\end{array}\right]=\left[\begin{array}{cc}
a+(b+c) & a+(b+c) \\
a+(b+c )& a+(b+c)
\end{array}\right]=(a)+((b)+(c))\end{equation}.

Did I correctly prove the associative axiom correctly?

Thanks
Cbarker1
 
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  • #2
Yes, your steps are correct (although there are a couple of redundant steps you could have skipped in the proof of associativity). Another way to prove the result is to show that the obvious mapping $f:\mathbb Z\to T;x\mapsto(x)$ is an isomorphism.
 

Related to Proving matrix group under addition for associative axiom

What is a matrix group under addition?

A matrix group under addition is a set of matrices that are closed under addition, meaning that when two matrices from the group are added together, the result is also a matrix within the group. This forms a mathematical structure that follows certain rules and properties.

What does it mean for a matrix group under addition to have an associative axiom?

The associative axiom for a matrix group under addition states that when three matrices are added together, the grouping of the operations does not affect the final result. In other words, (A + B) + C = A + (B + C) for all matrices A, B, and C in the group.

How is the associative axiom proven for a matrix group under addition?

The associative axiom for a matrix group under addition can be proven through a mathematical proof that shows that the grouping of operations does not change the final result. This is typically done by using the properties of matrices and the definition of a matrix group under addition.

Why is proving the associative axiom important for a matrix group under addition?

The associative axiom is important because it ensures that the matrix group under addition is a well-defined mathematical structure. It guarantees that the group follows certain rules and properties, making it easier to study and apply in various mathematical problems and applications.

What are some real-world examples of matrix groups under addition?

Matrix groups under addition can be found in various fields such as physics, engineering, and computer science. For example, in physics, matrices are used to represent transformations and operations on physical systems. In engineering, matrices are used for data analysis and modeling. In computer science, matrices are used for image processing and data compression. All of these applications involve matrices that form a group under addition.

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