Defining Contrast Matrix

#1
Suppose I have a binary Covariate \(X\) which is defined as

\( X =
\begin{cases}
1, & \text{if treatment group} \\
0, & \text{if placebo group}
\end{cases}\)

The model is \(\mathbb E[Y] = \beta_0 + \beta_1X,\)
where \(Y\) is a continuous random variable.

If \(X=0\), then \(\mathbb E[Y] = \beta_0\ldots (1)\),

if \(X=1\), then \(\mathbb E[Y] = \beta_0 + \beta_1\ldots (2)\).

Now I want to calculate the Wald Statistics \(W^2 = (\mathbf L\mathbf\beta)'\{\mathbf L\hat{\text{Cov}(\hat{\mathbf\beta})}\mathbf L'\}(\mathbf L\mathbf\beta),\)

where \(\mathbf L\) is a contrast matrix.

But I can't write the contrast matrix for both \((1)\) and \((2)\).

The hypothesis is \(\mathbf L\mathbf\beta = 0,\)

where \(\mathbf\beta =
\begin{pmatrix}
\beta_0\\
\beta_1\\
\end{pmatrix}.
\)

For \((1)\), I tried to write the contrast matrix \(\mathbf L=(1, 0)\) so that

\(\mathbf L\mathbf\beta = 0\)
\(\Rightarrow (1, 0)\begin{pmatrix}
\beta_0\\
\beta_1\\
\end{pmatrix}=0\)
\(\Rightarrow \beta_0 = 0,\)

but necessarily the contrast matrix is incorrect as the row sum of a contrast matrix is equal to \(0\). How can I define the contrast matrix?