Existence question about matrix and ranks.

i need to prove diprove that there exist matrices A1,A2,...,As such that rank Ai=1 for every i=1,...,s and A=A1+A2+...+As with rankA=10.
my feeling this is not true, i thought trying to prove this by ad absrudum, let us assume that they exist, then the rows of Ai are scalar multiple of one row vector, now im trying to show that if this is so then rankA cannot be equal to 10, but im stuck on that, can someone advise me on this problem?

 

nice example, yes it does work, and A=I_10.