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ptolema
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Homework Statement
Show that if A is any 3x3 matrix having rank 1, then there exist a 3x1 matrix B and a 1x3 matrix c such that A=BC
Homework Equations
rank (BC)=rank (A)=1
rank (BC) [tex]\leq[/tex] rank (B) and rank (BC) [tex]\leq[/tex] rank (C)
The Attempt at a Solution
I prove that if B is a 3x1 matrix and C is a 1x3 matrix, then the 3x3 matrix BC has rank at most 1 (rank BC [tex]\leq[/tex] 1) in a different part of the problem. I'm not sure if that would be useful in this proof, though; this one is more like proving the converse. This is how far I got with this proof in particular:
define A=BC,where B is 3×n matrix and C is n×3 matrix
rank BC=1 ≤ rank B ≤ n,1 ≤ rank B ≤ 3
1 ≤ rank C ≤ n,1 ≤ rank C ≤ 3
from here, I need to show that n=1, but I don't know how to get to that point. A little help here?