### Optimizing a function of a given matrix

Let us consider the $4\times 4$ symmetric matrix $$ A_x=\left(\begin{array}{rrrr}
0 & 1 & 1 & 1 \
1 & 0 & 2^x & 2^x \
1 & 2^x & 0 & 2^x \
1 & 2^x & 2^x & 0
\end{array}\right) $$

Here I need to find $\min { x>0: det(A_x)=0 \, or \, ||A_x^{-1}||=0 } ,$ where by $||M||$ we mean the sum of all entries of the matrix $M.$ I'm looking for a general sage program where my input will be a matrix with entries as functions of an inderminant (like the matrix $A_x$ above) which will give me the unique $x$ corresponding to my matrix. If no such real value exists, it should result as $\infty$ Can anyone help me? Thank you in advance.