Supposing $X$ is the general inverse of $A$, that $AXA = A$. Then $XA$ is idempotent, that is $(XA)(XA) = XA$.
Why is the rank of $XA$ equal to the rank of $A$ ?
Thanks.
2026-03-29 03:37:04.1774755424
The rank of general inverse of $A$ times $A$?
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On the one hand $\text{rank}(XA)\leq \min\left(\text{rank}(X), \text{rank}(A)\right)\leq \text{rank}(A)$.
On the other hand $\text{rank}(A)=\text{rank}(AXA)\leq \min\left(\text{rank}(A), \text{rank}(XA)\right)\leq \text{rank}(XA)$.