TITLE: Maximal dimension of subspace of matrices whose products have zero trace
QUESTION [7 upvotes]: In the space of all real matrices with dimension $n$, find the maximal possible dimension of a subspace $V$ such that $\forall X,Y\in V,\, \operatorname{tr}(XY)=0$.

REPLY [6 votes]: If $V$ is a subspace of real matrices of size $n$ with the property that $\operatorname{tr}(XY)=0$ for all $X,Y\in V$, then $\operatorname{tr}(X^2)=0$ for all $X\in V$. 
Let $W$ be a subspace of real matrices of size $n$ with the property that $\operatorname{tr}(X^2)=0$ for all $X\in W$. Then $\operatorname{tr}((X+Y)^2)=0$ for all $X,Y\in W$. But $\operatorname{tr}((X+Y)^2)=\operatorname{tr}(X^2)+\operatorname{tr}(Y^2)+2\operatorname{tr}(XY)$. It follows that $\operatorname{tr}(XY)=0$ for all $X,Y\in W$. 
This shows that the problem reduces to the following: Investigations about the trace form which is already solved.