Let S = {A = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd><mtd><mi>c</mi></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mi>a</mi></mtd><mtd><mi>d</mi></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mi>b</mi></mtd><mtd><mi>e</mi></mtd></mtr></mtable></mfenced></math> : a, b, c, d, e ∈ {0, 1} and |A| ∈ {-1, 1}}, where |A| denotes the determinant of A. Then the number of elements in S is ____.
Step-by-Step Solution
Key Concept: The determinant of the matrix A is given by |A| = 0(ae - bd) - 1(e - d) + c(b - a) = d - e + cb - ca. We need to find the number of tuples (a, b, c, d, e) with entries in {0, 1} such that d - e + c(b - a) \in {-1, 1}.
The determinant |A| = 0(ae - bd) - 1(e - d) + c(b - a) = d - e + c(b - a). Given a, b, c, d, e \in {0, 1}, we need |d - e + c(b - a)| = 1. Case 1: c = 0. Then |d - e| = 1, which implies (d, e) \in {(0, 1), (1, 0)}. Since a, b can be any of the 2 values, there are 2 * 2 * 2 = 8 such matrices. Case 2: c = 1. Then |d - e + b - a| = 1. Let x = d - e and y = b - a. x, y \in {-1, 0, 1}. We need |x + y| = 1. Possible pairs (x, y) are (1, 0), (0, 1), (-1, 0), (0, -1), (1, -1), (-1, 1). Counting the combinations of (d, e) and (b, a) for each: (d, e) can be (0,0), (0,1), (1,0), (1,1) giving x = 0, -1, 1, 0. (b, a) can be (0,0), (0,1), (1,0), (1,1) giving y = 0, -1, 1, 0. Total combinations for (x, y) such that |x+y|=1 is 8. Total elements = 8 + 8 = 16.
Correct Answer: 4