Understand the rank of $ \begin{bmatrix} A&b\\ b^{*}&0 \end{bmatrix}$ 
Let $A \in M_n(C)$ and $b$ be a column vector of n complex complements. Denote $\widetilde A = \begin{bmatrix}
A&b\\
b^{*}&0 
\end{bmatrix} $ If $rank(\widetilde A)=rank(A)$, which of the following is true? 
(a) $Ax = b$ has infinitely many solutions. 
(b) $Ax = b$ has a unique solution.
(c)  $\widetilde Ax = 0$ has only solution x = 0. 
(d)  $\widetilde Ax = 0$ has nonzero solutions.
Zhang, Fuzhen. Linear Algebra 

I am assuming that $b^*$ is the conjugate transpose. Could you help me construct an example $A$ and $\widetilde A$ such as $rank(\widetilde A)=rank(A)$ ? I have hard time doing that without assuming that $b=0$.

Hint: If both matrices have the same rank (column rank = row rank), then $b$ lies in the column space of $A$ and so there is a linear combination of the columns of $A$ which gives $b$.
