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It may take some reordering of the rows (and correspondingly the columns) to make $A_1$ nonsingular.

If it is, consider $$A ( A+I) \pmatrix{A_1^{-1} & -A_1^{-1} B\cr 0 & I\cr} = \pmatrix{I & 0\cr B^T A_1^{-1} & A_2 - B^T A_1^{-1} B\cr}$$

If this has rank $k$, the lower right block must be $0$.

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It may take some reordering of the rows (and correspondingly the columns) to make $A_1$ nonsingular.

If it is, consider $$A \pmatrix{A_1^{-1} & -A_1^{-1} B\cr 0 & I\cr} = \pmatrix{I & 0\cr B^T A_1^{-1} & A_2 - B^T A_1^{-1} B\cr}$$

If this has rank $k$, the lower right block must be $0$.