Matrix of a relation on a set

Question

If I have a Matrix $A=\begin{bmatrix} 0&0&0\\ 0 & 0 & 0 \\0&0&0\end{bmatrix}$ why is this both symmetric and anti-symmetric? If I had a Matrix $B=\begin{bmatrix} 1&1&1\\1&1&1\\1&1&1\end{bmatrix}$ would this also be symmetric and anti-symmetric?

Answer 1

The first matrix represents the empty relation on a $3$-element set. So both symmetry and anti-symmetry are satisfied vacuously.

Now $B$ represents the relation on a $3$-element set where any two elements are related. Hence, it is symmetric, but not anti-symmetric (e.g $1 \sim 2$ and $2 \sim 1$, but $1 \ne 2$ )