Portfolio choice problem of a CARA investor with n risky assets
The problem is equivalent to maximizing

$$
\max_\phi \left( \phi ' \mu - \frac{1}{2} \alpha \phi ' \Sigma \phi \right)
$$

subject to

$$
\mathbf{1}' \phi = w_0.
$$

(boldface 1 is column vector of ones, ' stands for transposition).

The lagrangian is

$$
L(\phi, \lambda) = \phi ' \mu - \frac{1}{2} \alpha \phi ' \Sigma \phi - \lambda \left( \mathbf{1}' \phi - w_0 \right),
$$

and its jacobian wrt. $\phi$ is

$$
\mathrm{D}_\phi L(\phi, \lambda) = \mu' - \alpha \phi ' \Sigma - \lambda \mathbf{1}'.
$$

First order conditions require that the jacobian is equal to zero (in each element), plus the original budget constraint, which together (after transposing the jacobian) yields a system of linear equations in $(\phi,\lambda)$:

$$
\begin{split}
\Sigma \phi + \mathbf{1} \lambda &amp;= \frac{1}{\alpha}\mu \\
\mathbf{1}' \phi &amp;= w_0
\end{split}
$$

We can solve for $\phi$ as a function of multiplier $\lambda$:

$$
\phi = \frac{1}{\alpha} \Sigma^{-1} \mu - \lambda \Sigma^{-1} \mathbf{1} 
$$

Plugging this into the budget constraint and some algebra yields an expression for $\lambda$:

$$
\lambda = \frac{\frac{1}{\alpha}\mathbf{1}' \Sigma^{-1} \mu - w_0}{\mathbf{1}' \Sigma^{-1} \mathbf{1}},
$$

and substituting this into the expression for $\phi$ should yield the result as given in the question.