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#1 2012-11-07 18:45:03

princess snowwhite
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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.

#2 2012-11-07 22:38:27

scientia
Full Member

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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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