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!

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.