# 2x2 matrices with sum of diagonal entries equal zero

Is the set of 2x2 matrices with sum of diagonal entries equal zero a vector space over the real numbers under the usual matrix matrix addition and scalar matrix multiplication?

$$\left\{ \left. \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \right| \, a_{11} + a_{22} = 0 \right\}$$
and so it is the set of solutions to a linear homogeneous equation and thus a vector space of dimension $4 - 1 = 3$.
Set of all matricess of below type with a,b,c $\in \mathbb{R}$ \begin{bmatrix} a & b\\ c & -a\\ \end{bmatrix} are the $2 \times 2$ matrices of trace zero form a vector subspace of $M(2,\mathbb{R})$ with dimension 3. Since, there are three free arbitrary constants(a,b,c). Vector addition and scalar multiplication closed under usual matrix addition and scalar with matrix multiplication.