Let x be a non-zero real number. A determinant is chosen from the set of all determinants of order 2 with entries x or –x
only. The probability that the value of the determinant is non-zero is

determinants are

$$\begin{gathered} \left|\begin{array}{cc}\mathrm{x}& \mathrm{x}\\ \mathrm{x}& \mathrm{x}\end{array}\right| \left|\begin{array}{cc}\mathrm{x}& \mathrm{x}\\ \mathrm{x}& -\mathrm{x}\end{array}\right| \left|\begin{array}{cc}\mathrm{x}& \mathrm{x}\\ -\mathrm{x}& \mathrm{x}\end{array}\right| \left|\begin{array}{cc}\mathrm{x}& -\mathrm{x}\\ \mathrm{x}& \mathrm{x}\end{array}\right| \left|\begin{array}{cc}-\mathrm{x}& \mathrm{x}\\ \mathrm{x}& \mathrm{x}\end{array}\right| \left|\begin{array}{cc}\mathrm{x}& -\mathrm{x}\\ \mathrm{x}& -\mathrm{x}\end{array}\right| \\ \left|\begin{array}{cc}-\mathrm{x}& \mathrm{x}\\ -\mathrm{x}& \mathrm{x}\end{array}\right| \left|\begin{array}{cc}\mathrm{x}& \mathrm{x}\\ -\mathrm{x}& -\mathrm{x}\end{array}\right| \left|\begin{array}{cc}-\mathrm{x}& -\mathrm{x}\\ \mathrm{x}& \mathrm{x}\end{array}\right| \left|\begin{array}{cc}\mathrm{x}& -\mathrm{x}\\ -\mathrm{x}& \mathrm{x}\end{array}\right| \left|\begin{array}{cc}-\mathrm{x}& \mathrm{x}\\ \mathrm{x}& -\mathrm{x}\end{array}\right| \left|\begin{array}{cc}\mathrm{x}& -\mathrm{x}\\ -\mathrm{x}& -\mathrm{x}\end{array}\right| \\ \left|\begin{array}{cc}-\mathrm{x}& \mathrm{x}\\ -\mathrm{x}& -\mathrm{x}\end{array}\right| \left|\begin{array}{cc}-\mathrm{x}& -\mathrm{x}\\ \mathrm{x}& -\mathrm{x}\end{array}\right| \left|\begin{array}{cc}-& -\mathrm{x}\\ -\mathrm{x}& \mathrm{x}\end{array}\right| \left|\begin{array}{cc}-\mathrm{x}& -\mathrm{x}\\ -\mathrm{x}& -\mathrm{x}\end{array}\right| \end{gathered}$$

$\mathrm{n}\left(\mathrm{S}\right)=16$    E = det value is non zero

$\mathrm{n}\left(\mathrm{E}\right)=8 \Rightarrow \mathrm{P}\left(\mathrm{E}\right)=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$