Let $$A=\begin{pmatrix} 0&0&1&0\\ 1&0&0&1\\ 0&-1&3&0\\ 2&1&4&-3 \end{pmatrix}$$ and $$B=\begin{pmatrix} 1&2&3&4&5&6\\ 2&3&4&5&6&7\\ 3&4&5&6&7&8\\ 4&5&6&7&8&9 \end{pmatrix}.$$
I was intended to find the nullity of the matrix $B^T A$, where $B^T$ denotes the transpose of $B$. One method is to compute $B^T A$ and then find the null space of the product. I'm sure it will work, and no subtle trick is involved, but the algebra is really arduous. I was wondering if there is any property that can significantly reduce the amount of computation. Thank you.
Edit. I noticed that $A$ has full rank, and I know the rank-nullity theorem. Does it help any?
You can do Gram Schmidt process and count the number of zero vectors.