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The problem given asks us to determine if the set \( G \) of all \( 2 \times 2 \) matrices of the form 
\[
\begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix}
\]
where \( a, b \in \mathbb{R} \), forms a group under matrix addition. Let's analyze this by checking the group properties.

### Group Properties under Matrix Addition

To determine if \( G \) is a group under matrix addition, we need to check the following properties:

1. **Closure**: If we add any two matrices in \( G \), the result should also be a matrix in \( G \).
   
2. **Associativity**: Matrix addition is associative, so this property holds for any set of matrices.

3. **Identity Element**: There should exist an element \( E \) in \( G \) such that for any matrix \( A \in G \), \( A + E = A \).

4. **Inverse Element**: For every matrix \( A \in G \), there should exist a matrix \( B \in G \) such that \( A + B = E \), where \( E \) is the identity element.

Let's go through each property one by one.

#### 1. Closure
If we take two matrices \( A = \begin{bmatrix} a_1 & 0 \\ 0 & b_1 \end{bmatrix} \) and \( B = \begin{bmatrix} a_2 & 0 \\ 0 & b_2 \end{bmatrix} \) from \( G \), their sum is
\[
A + B = \begin{bmatrix} a_1 + a_2 & 0 \\ 0 & b_1 + b_2 \end{bmatrix}.
\]
Since \( a_1 + a_2 \) and \( b_1 + b_2 \) are real numbers, \( A + B \) is also of the form \( \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix} \), where \( a, b \in \mathbb{R} \). Therefore, \( G \) is closed under matrix addition.

#### 2. Associativity
Matrix addition is associative in general, so this property holds for \( G \) as well.

#### 3. Identity Element
The identity element under matrix addition is the zero matrix:
\[
E = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}.
\]
This matrix is in \( G \) (since \( a = 0 \) and \( b = 0 \) are real numbers), and for any \( A = \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix} \) in \( G \),
\[
A + E = \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix}.
\]
Thus, the identity element exists in \( G \).

#### 4. Inverse Element
For any \( A = \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix} \) in \( G \), the additive inverse is \( -A = \begin{bmatrix} -a & 0 \\ 0 & -b \end{bmatrix} \), which is also of the form \( \begin{bmatrix} a & 0 \\ 0 & b \end{bmatrix} \) where \( a, b \in \mathbb{R} \). This matrix \( -A \) is in \( G \), and it satisfies \( A + (-A) = E \).

### Conclusion
Since \( G \) satisfies closure, associativity, the existence of an identity element, and the existence of inverses under matrix addition, \( G \) is indeed a group under matrix addition.

Let me know if you need further details or have any questions!

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### Related Questions:
1. Can you verify if \( G \) is a group under matrix multiplication instead of addition?
2. What changes in the properties if we restrict \( a \) and \( b \) to be positive real numbers?
3. Is the set of all diagonal matrices a group under matrix addition?
4. How does matrix addition differ from matrix multiplication in terms of group properties?
5. What if we consider \( a \) and \( b \) as integers? Would \( G \) still be a group?

### Tip:
For groups under addition, always verify each property explicitly, even if they seem obvious, as it helps solidify understanding of the structure.

Let G be the set of all 2x2 matrices of the form [[a, 0], [0, b]], where a, b ∈ R. Is G a group under matrix addition? Prove your answer.

Explore whether the set G of 2x2 diagonal matrices of the form [[a, 0], [0, b]] (where a, b are real numbers) forms a group under matrix addition. Understand the group properties and proofs required in linear algebra and group theory. Suitable for undergraduate students.

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