If a, b, c > 0 and x, y, z ∈ R, then the determinant <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="|
Step-by-Step Solution
Key Concept: Use the identity (u+v)^2 - (u-v)^2 = 4uv. Here, (a^x + a^-x)^2 - (a^x - a^-x)^2 = 4(a^x)(a^-x) = 4. Applying C1 -> C1 - C2 makes the first column all 4s, and the determinant becomes 0.
Let the determinant be \Delta. Applying C1 \to C1 - C2, we get (a^x + a^-x)^2 - (a^x - a^-x)^2 = 4, (b^y + b^-y)^2 - (b^y - b^-y)^2 = 4, (c^z + c^-z)^2 - (c^z - c^-z)^2 = 4. Thus, the first column becomes [4, 4, 4]^T. Since the first column is a constant multiple of [1, 1, 1]^T, and the third column is [1, 1, 1]^T, the determinant is 0.
Correct Answer: D