A porfolio $w=(w_1,...,w_n) \in \mathbb{R}^n$.
The expected return of the porfolio is: $R(w)=E(\sum_{i=1}^n w_iX_i)=\sum_{i=1}^n w_ir_i$
The variance $V(w)=w^\top\Sigma w$
Optimization problem: $minimize \ V(w) \ subject \ to \ R(w)=\mu, \sum_{i=1}^n w_i = a$
The text book mention that the unique solution is $w^*=\sum^{-1} \begin{pmatrix}| & | \\ \mathbb{1}&r\\ |&| \end{pmatrix} A^{-1} \begin{pmatrix} a\\ \mu \end{pmatrix}$,
where $A=\begin{pmatrix} \mathbb{1}^\top \Sigma^{-1} \mathbb{1}& \mathbb{1}^\top \Sigma^{-1} r\\ \mathbb{1}^\top \Sigma^{-1} r & r^\top \Sigma^{-1} r \end{pmatrix} $, whhich is invertible. Assume $r \notin Span(1), \mathbb{1}$ denotes the all-ones vector in dimension $n$
How to show $A$ is invertible and how to derive the unique solution $w^*$