I have a followup question on this topic, i.e. adding covariates of no interest in the design, such as age and sex.
I have a similar situation as what explained above:

P1T1 1 0 1 0
P2T1 0 1 1 0
P1T2 1 0 0 1
P2T2 0 1 0 1
P1T3 1 0 0 0
P2T3 0 1 0 0

Echange block: [1 2 1 2 1 2]
Contrast F test [0 0 1 1]

Now I would like to add age and sex as between subjects covariates. If I do as a design matrix:

P1T1 1 0 1 0 20 0
P2T1 0 1 1 0 30 1
P1T2 1 0 0 1 20 0
P2T2 0 1 0 1 30 1
P1T3 1 0 0 0 20 0
P2T3 0 1 0 0 30 1

Echange block: [1 2 1 2 1 2]
Contrast F test [0 0 1 1 0 0]

I get an error:
Warning: Rank deficient, rank = 839, tol = 1.242192e-09.

My question is, is it possible to have a between-subject covariate in such a design?
Thank you very much.