Linear Algebra by Friedberg Theorem 4.2: If the rank of matrix is two, then the first column can't be 0

Question

In the proof of Theorem 4.2 from Linear Algebra by Friedberg, it mentions that if the rank of $$ A = \pmatrix{A_{11} & A_{12} \\ A_{21} & A_{22}} $$ is $2$, we must have $A_{11} \neq 0$ or $A_{21} \neq 0$. I understand that this is because $A$ must have two linearly independent columns/rows (correct me if I am wrong), but why doesn't it say that $A_{12} \neq 0$ or $A_{22} \neq 0$?

Answer 1

If either $A_{11} = A_{21} = 0$ or $A_{12} = A_{22} = 0$, then the rank is less than 2 (Note that this is not an if and only if).

Both are true, and they decided to mention only the former.