I have to show that

If $A$ and $B$ be two matrices of the same order $n$, then $$ \text{rank}\, A + \text{rank}\,B \le\text{rank}\,AB + n $$

In a previous exercise I have already shown that

$$ \text{dim}\,(\text{ker}\,(AB)) \le \text{dim}\,(\text{ker}\,(A)) + \text{dim}\,(\text{ker}\,(B)) $$

Is it enough to use the rank-nullity theorem and subbing in

$$ \text{dim}\,(\text{ker}\,(AB))= \text{dim}\, (V) - \text{rank}\,(AB) $$

to show that the two inequalities are basically saying the same thing?


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