$A_1\cdots A_s=0$, prove $\sum_{i=1}^s \operatorname{rank}(A_i) \leq (s-1)n$

Question

$A_1,\ldots,A_s$ are $n\times n$ matrices which satisfy $A_1\cdots A_s=0$

Want to prove

$$\sum_{i=1}^s \operatorname{rank}(A_i) \leq (s-1)n$$

I have no idea how to add up the ranks of matrices, so I didn't succeed by induction. Any help will be appreciated!

Answer 1

for s=2,it is clear. If $s>2$,you can show this by prove $rank(AB)+n≥rank(A)+rankB.$ you can prove this by elementary transformation for a matrix $[AB,0;0,I]$.then by induction,you can get the result.