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#IISc2012Research Rank of a matrix, a (i,j)=0,if i+j odd, else 1
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Each row of the matrix is either of the form 1,0,1,0,1,..., or of the form 0,1,0,1,0,1,...

These two types of rows are independent and hence rank of matrix is 2.

As a special case, If matrix is of size 1*1, then rank would be 1 (Thanks Suraj for pointing it out).
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We should also consider 1X1 matrix. Therefore rank = min(n,2)
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rank must be 2..... 3X3 matrix will be

1 0 1    and it can be transfer into by applying   1 0 0

0 1 0      r3=r3-r1    and c3=c3-c1                        0 1 0

1 0 1                                                                     0 0 0