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.

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.]]>