<p>A determinant is chosen at random from the set of all \(2 \times 2\) determinants with entries from \(\{0, 1\}\). The probability that the value of the chosen determinant is positive is</p>
Step-by-Step Solution
Key Concept: Total 2 \times 2 matrices with entries from {0,1}: 2^4 = 16. det > 0 requires ad - bc > 0, i.e., ad = 1 and bc = 0.
<p>Total matrices: $2^4 = 16$.</p><p>$\det = ad - bc > 0 \Rightarrow ad = 1, bc = 0$.</p><p>$ad = 1$: $a = d = 1$ (1 way). $bc = 0$: $(b,c) \in \{(0,0),(0,1),(1,0)\}$ (3 ways).</p><p>Favorable $= 1 \times 3 = 3$.</p><p>$P = \dfrac{3}{16}$</p>
Correct Answer: B