Let $A = \begin{bmatrix} A\_{11} & A\_{12} \\ A\_{21} & A\_{22} \end{bmatrix}$ be an invertible matrix where $A\_{11}$ is square. Let $A^{-1} =: B = \begin{bmatrix} B\_{11} & B\_{12} \\ B\_{21} & B\_{22} \end{bmatrix}$ and shape of $B\_{11}$ is same as $A\_{11}$.


What is known about the relationships between the ranks of $A\_{21}$ and $B\_{21}$? For example, if either $A\_{11}$ or $A\_{22}$ is invertible, then it can be easily shown using the block inverse formula that the ranks are equal. I presume when this is not true, the equality does not hold in general. Is this correct? In that case, what bounds are known?



Rank of a block of an invertible matrix
That paper from Strang and Nguyen gives the answer: <https://epubs.siam.org/doi/abs/10.1137/S0036144503434381>


See "theorem 2.1: (nullity theorem). Complementary submatrices of a square matrix and its inverse have the same nullity". In your case the complementary submatrices are exactly $A\_{21}$ and $B\_{21}$, so their ranks are equal.


