From AB=0, the column vectors of matrix B are all solution vectors of homogeneous linear equations Ax=0.

And B≠0, so the homogeneous linear equation system Ax=0 has a non-zero solution, so |A|=0, k=4.

(If we use |A||B|=|AB|=0, we know B≠0, but we don't say |B|≠0, so we can't deduce |A|=0).