Two orders on projections in $M_2(\mathbb{F}_3)$

Question

Let us consider the space $M_2(\mathbb{F}_3)$ consists of $2\times 2$ matrices over the field $\mathbb{F}_3=\{0,1,2\}$.

1- For a given matrix $A$, we denote its transpose by $A^t$.

2- We say $A$ is positive if there exists finitely many matrices $B_1,\cdots, B_n$ with $A=\sum B_iB_i^t$

3- By a projection $P$, we mean $P=P^t=P^2$.

Question. Do there exist projections $P$ and $Q$ in $M_2(\mathbb{F}_3)$ such that $Q-P$ is a positive matrix and $PQ\neq P$.

Answer 1

Clearly $Q=diag(1,0),P=diag(0,1)$ are projections.

Thus $Q-P=diag(1,2)=QQ^T+PP^T+PP^T$ is a positive matrix.

Last point: $PQ=0\not= P$.