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 I am trying to show that if this is so then rankA cannot be equal to 10, but I am 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.