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### Problem 13

Let $H$ be the subset of $M_{2}(\mathbb{R})$ consisting of all matrices of the form $\begin{bmatrix} a & -b \\ b&a \end{bmatrix}$ for $a,b \in \mathbb{R}$. Is $H$ closed under matrix addition? Matrix multiplication?

**Solution**

Let $\begin{bmatrix} a & -b \\ b&a \end{bmatrix}, \begin{bmatrix} c & -d \\ d&c \end{bmatrix}$ be in $H$ where $a, b, c, d \in \mathbb{R}$. Now for matrix addition,

$\begin{bmatrix} a & -b \\ b&a \end{bmatrix} + \begin{bmatrix} c & -d \\ d&c \end{bmatrix} = \begin{bmatrix} a+c & -b-d \\ b+d&a+c \end{bmatrix}$, where all entries are real numbers in the sum by properties of addition in $\mathbb{R}$. Hence, closed under matrix addition.

For matrix multiplication,

$\begin{bmatrix} a & -b \\ b&a \end{bmatrix} \cdot \begin{bmatrix} c & -d \\ d&c \end{bmatrix} = \begin{bmatrix} ac-bd & -ad-bc \\ bc+ad&-bd+ac \end{bmatrix}$, where all entries are real numbers in the product by properties of addition and multiplication in $\mathbb{R}$. Hence, closed under matrix multiplication.