Suppose we are given three real matrices $A$,$B$,$C$ of sizes $40 × 20, 20×30, 30×40$. And the rank of the matrix $AB$, $BC$, $C$ are $11,12,15$ respectively. Then how should I find minimum possible rank of matrix $ABC$?
I tried using formula $rank(AB) \leq min(rank(A),rank(B))$
As given $rank(B)=15$ and $rank(BC)=12$ $\implies rank(BC)\leq (rank(B),15)$ $\implies rank(B) = 12$
Similarly, $rank(A) = 11$
Now I stuck how to find minimum $rank(ABC)$ because the formula gives the maximum possible rank!