Let's take a simpler example:

Code:

df <- data.frame(Sample=factor(c(1:10)), Group=c(rep("A",5), rep("B",5)))
mm <- model.matrix(~Sample+Group, df)
mm

Have a look at "mm". The last column is simply the sum of columns 6-10, meaning that if you estimate those coefficients, then the last coefficient is completely determined by them (or inversely, if you estimate it then the others are already determined). Something along these lines is also the case for the model matrix in your design.