The question asks to determine whether the set $S$, consisting of all symmetric $2 \times 2$ matrices with real entries, is a subspace of $\mathbb{R}^{2 \times 2}$ under usual matrix addition and scalar multiplication. To solve this, we need to check if $S$ satisfies the three subspace criteria:

Subspace Criteria

1. Zero Matrix: The set $S$ must contain the zero matrix.
2. Closed under Addition: If $A$ and $B$ are both in $S$, then their sum $A + B$ must also be in $S$.
3. Closed under Scalar Multiplication: If $A \in S$ and $c$ is any scalar, then $cA$ must also be in $S$.

1. Zero Matrix:

The zero matrix for $2 \times 2$ matrices is: $\mathbf{0} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$ This matrix is symmetric (since $\mathbf{0}^T = \mathbf{0}$), so it belongs to $S$. Hence, $S$ contains the zero matrix.

2. Closed under Addition:

Let $A = \begin{pmatrix} a & b \\ b & d \end{pmatrix}$ and $B = \begin{pmatrix} e & f \\ f & g \end{pmatrix}$ be two symmetric $2 \times 2$ matrices in $S$. Their sum is: $A + B = \begin{pmatrix} a + e & b + f \\ b + f & d + g \end{pmatrix}$ Since $(A + B)_{12} = (A + B)_{21}$ (the off-diagonal elements are equal), the matrix $A + B$ is symmetric. Therefore, $S$ is closed under addition.

3. Closed under Scalar Multiplication:

Let $A = \begin{pmatrix} a & b \\ b & d \end{pmatrix} \in S$, and let $c$ be a scalar. Then: $cA = \begin{pmatrix} ca & cb \\ cb & cd \end{pmatrix}$ Since $(cA)_{12} = (cA)_{21}$, the matrix $cA$ is symmetric. Therefore, $S$ is closed under scalar multiplication.

Conclusion:

Since the set $S$ contains the zero matrix, is closed under addition, and is closed under scalar multiplication, it satisfies all the conditions to be a subspace of $\mathbb{R}^{2 \times 2}$. Hence, $S$ is a subspace of $\mathbb{R}^{2 \times 2}$.

Would you like more details or have any further questions? Here are some related questions you might consider:

1. What are the dimensions of the subspace $S$ of symmetric matrices in $\mathbb{R}^{2 \times 2}$?
2. How can we generalize this concept for symmetric $n \times n$ matrices?
3. Can we find a basis for the set of symmetric $2 \times 2$ matrices?
4. How would the properties change if we considered skew-symmetric matrices instead?
5. How does the subspace of symmetric matrices compare to the subspace of diagonal matrices?

Tip: A matrix is symmetric if and only if it is equal to its transpose!