Existence question about matrix and ranks.

[MathematicalPhysicist]

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?

[HallsofIvy]

What about about 10 10 by 10 matrices A1 with 1 in the first row, first column 0 everywhere else, A2 with 1 in the second row, second column, 0 everywhere else, A3 with 1 in the third row, third column, 0 everywhere else, etc. What is the rank of each of thosef? What is the rank of their sum?

[MathematicalPhysicist]

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