1.
The total number of matrices
A=(02y12xy12xy1),(x,yR,  xy0)A = \begin{pmatrix} 0 & 2y & 1 \\ 2x & y & -1 \\ 2x & -y & 1 \end{pmatrix}, \quad (x, y \in \mathbb{R}, \; xy \neq 0)

for which ATA=3I3A^{T} A = 3 I_{3} is:
Matrices and Determinants - Medium - Question