Checkerboard the Matrix
MATLAB - 114 129 bytes
function n=c(m)
n=rref(m);l=size(n,1);for i=1:l
e=1.5+.5*(-1)^i:2:l;N=n(e,:);n(e,:)=N+repmat(n(i,:),length(e),1).*max(N).^2;end

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For a given nxm matrix, there are three forms of the rref of the matrix.

The identity matrix (with optional all-zero columns) (m&gt;=n)
Full rank matrix where columns that do not contain a row-leading 1 can have any value (m&gt;n)
Identity matrix with zero rows on the bottom (m&lt;n)

1 and 3 are easily made into a checkerboard by looping through each row and adding every odd/even row (which would have the 1s in the appropriate places)
for 2, however, it is impossible to make a checkerboard matrix unless the matrix is already a checkerboard. This is because the columns that have nonzero values can never be reduced because there is no column with a row-leading 1, and any ERO would add a value in another space that would ruin the checkerboard pattern. I'm certain there's a more rigorous explanation but linear algebra is not easy.