 # If the rank of A=((A_(11),A_(21)),(A_(12),A_(22)) is 2, we must have A_(11) != 0 or A_(21) != 0. I understand that this is because A must have two linearly independent columns/rows , but why doesn't it say that A_(12) != 0 or A_(22) != 0? themediamafia73 2022-09-11 Answered
If the rank of
$A=\left(\begin{array}{cc}{A}_{11}& {A}_{12}\\ {A}_{21}& {A}_{22}\end{array}\right)$
is 2, we must have ${A}_{11}\ne 0$ or ${A}_{21}\ne 0$. I understand that this is because A must have two linearly independent columns/rows , but why doesn't it say that ${A}_{12}\ne 0$ or ${A}_{22}\ne 0$?
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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.