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#1 2012-11-06 19:45:03

princess snowwhite
Member
Registered: 2012-11-06
Posts: 29

Rank Of Matrix

I HAVE TWO QUESTIONS-
1>  If A be a non-zero column matrix and B be a non-zero row matrix, then
show that the rank of AB is 1.
2>  If A and B be square matrices both of order n and of ranks p and q respectively, then
show that the rank of (AB) ≥ p + q - n.

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#2 2012-11-06 23:38:27

scientia
Member
Registered: 2009-11-13
Posts: 222

Re: Rank Of Matrix

(1)


Suppose
and
.

Then

is an
matrix. For
, the
th column of
is
and the
th column of
is
. Hence any two columns of
are linearly dependent. As
is a nonzero matrix, it follows that its rank is 1.

(2)
This is harder to prove. It is called Sylvester's inequality: http://www.artofproblemsolving.com/Foru … hp?t=43691.

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