SOLUTION: Consider the n × n matrix A where a(ij) [ 1 if i+j is even, [ 0 if i+j is odd. Find the rank and nullity of A.

Question 1167395: Consider the n × n matrix A where
a(ij) [ 1 if i+j is even,
[ 0 if i+j is odd.
Find the rank and nullity of A.

Answer by ikleyn(52781) (Show Source): You can put this solution on YOUR website!
.


They are chess-board pattern matrices, filled by 1 and 0.


Regarding the rank of such a matrices, notice that they have, actually, only 2 different lines - the first and the second, 


while the rest of the lines repeat either the first or the second lines.


Therefore, by using elementary linear transformations, they can be reduced to matrices having only two upper lines different,

while all other lines are zeroed.


Elementary matrix transformations do not change the range.


Therefore, the range is  2  and the null space has the dimension (n-2).



// The fact that the range is 2 means that the image of the linear operator with such a matrix has the dimension of 2.

Solved.

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