Let $S$ be the set of all possible values of the determinant of a matrix $M=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, where $a,b,c,d$ are chosen independently and uniformly from $\{0,1\}$. Let $X=\det(M)$. Consider the quadratic $t^2-\gamma t+\delta=0$, where $\gamma$ and $\delta$ are chosen independently from $S$ with probabilities proportional to their frequency in $M$. The probability that the roots are non-real complex is:
Step-by-Step Solution
Key Concept: $\det(M)\in\{-1,0,1\}$. $P(\det=1)=3/16$, $P(\det=0)=10/16$, $P(\det=-1)=3/16$. So $S=\{-1,0,1\}$ with frequencies $\{3,10,3\}$. For non-real roots: discriminant $\gamma^2-4\delta<0$, i.e., $\gamma^2<4\delta$. Only possible if $\delta=1$ (since $\delta\in\{-1,0,1\}$) and $\gamma\in\{-1,0,1\}$.
Since probabilities are proportional to frequencies: $P(X=-1)=3/16$, $P(X=0)=10/16$, $P(X=1)=3/16$. For non-real: need $\gamma^2<4\delta$, so $\delta=1$ and any $\gamma$. $P=\frac{3}{16}\times\frac{16}{16}=\frac{3}{16}=\frac{48}{256}$.
Correct Answer: D