number of matrices with a single nullspace point

Question

Suppose I fix a vector $v\in \mathbb{R}^n$. I want to count the number of maximal rank (in this case $n-1$) linearly independent matrices $A_1,\ldots,A_m$ for which $v$ lies in the nullspace of $all$ the $A_i$s, i.e., $A_iv=0$. What is the largest $m$ for which these set of matrices can exist?

Answer 1

If $Av=0, Bv=0$ then clearly $(A+B)v = Av+Bv=0$ and for any constant $k$, $(kA)v = 0$. This shows that the number of such matrices is infinite. Better question would be: How many Linearly independent matrices $A$ exist such that $Av=0$ for a fixed vector $v$?