Rank of a matrix less than its dimension

Question

Suppose I have matrices $A,B$ both of dimension, $2m$ by $2$. Then $$AB^TBA^T$$ is of dimension $2m$ by $2m$, $A^T$ denotes the transpose of $A$. I know that rank would mean the number of linearly independent vectors but what makes me confused is when my teacher said that if the rank of $A$ is 1 then the product of the matrices above is equal to the zero matrix.

Answer 1

Let $m=1$ and let $A = \begin{bmatrix}1&0\\1&0\end{bmatrix}$ and $B = \begin{bmatrix}1&0\\0&1\end{bmatrix}$. Then $A$ has rank $1$, but $$AB^\top B A^\top = AA^\top = \begin{bmatrix}1&1\\1&1\end{bmatrix}$$ is not the zero matrix.