Let S = \left\{ A = \begin{pmatrix} 0 & 1 & c \\ 1 & a & d \\ 1 & b & e \end{pmatrix} : a, b, c, d, e \in \{0, 1\} \text{ and } |A| \in \{-1, 1\} \right\}, \text{ where } |A| \text{ denotes the determinant of } A. \text{ Then the number of elements in } S \text{ is } \_\_\_\_.
Step-by-Step Solution
Key Concept: Calculate the determinant of the matrix A in terms of a, b, c, d, e and set it equal to 1 or -1. Since a, b, c, d, e are in {0, 1}, count the number of valid tuples (a, b, c, d, e) that satisfy the condition.
The determinant |A| = 0(ae - bd) - 1(e - d) + c(b - a) = d - e + cb - ca = d - e + c(b - a). Given a, b, c, d, e \in {0, 1}, we need |d - e + c(b - a)| = 1. By testing combinations, we find the number of such tuples is 4.
Correct Answer: 4