Given a linear model $Y = X\beta + \epsilon$ with three treatments and six subjects where $X$ is the design matrix, suppose $X = \begin{matrix}1 & 1 & 0\\ 1 & 1 & 0\\ 1 & 0 & 1\\ 1 & 0 & 1\\ 1 & -1 & -1\\ 1 & -1 & -1 \end{matrix}$ and
X'= \begin{matrix}1 & 0 & 0\\ 1 & 0 & 0\\ 1 & 1 & 0\\ 1 & 1 & 0\\ 1 & 0 & 1\\ 1 & 0 & 1 \end{matrix}
with response vector $Y=[Y_{11} Y_{12} Y_{21} Y_{22} Y_{31} Y_{32}]^{T}$
Do these two matrices give you the same model?
I've had some trouble trying to figure out how, given that $\beta = (\beta_1 \beta_2 \beta_3)^T$, these two designs can give the equivalent model. Don't they have entirely different values of $\beta$ for each $Y_{ij}$?
EDIT: The constraints are $\sum_{i=1}^{3}\beta_i = 0$