[QUE/VS-02009]

PRO/VS-2009

Consider the set of all vectors \(3\times3\) real matrices \(A\) for which

1. \(\text{Tr}(A)=0\)
2. \(\det A=0\)
3. \(A_{11}>0\)
4. either \(A_{11}, A_{22}, A_{33}\), all equal to zero
5. \(A^{\text{T}}=A\)
6. \(A^\text{T}=-A\)

 Give the dimensions of the vector spaces wherever appropriate and give a possible basis?