Matrices & Determinants
Properties of determinants
Grade Class 12
Question:
If a, b, c > 0 and x, y, z ∈ R, then the determinant <br> <img src="https://latex.codecogs.com/svg.latex?\begin{vmatrix} (a^x + a^{-x})^2 & (a^x - a^{-x})^2 & 1 \\ (b^y + b^{-y})^2 & (b^y - b^{-y})^2 & 1 \\ (c^z + c^{-z})^2 & (c^z - c^{-z})^2 & 1 \end{vmatrix}"/> is equal to -
(A) a^x b^y c^z
(B) a^{-x} b^{-y} c^{-z}
(C) a^{2x} b^{2y} c^{2z}
(D) zero
Step-by-Step Solution
Key Concept: Use the identity (u+v)^2 - (u-v)^2 = 4uv. Apply column operation C1 -> C1 - C2.
Let the determinant be \Delta. Using the property (u+v)^2 - (u-v)^2 = 4uv, we perform the operation C1 -> C1 - C2. The first column becomes: (a^x + a^-x)^2 - (a^x - a^-x)^2 = 4(a^x)(a^-x) = 4. Similarly, the other elements in the first column become 4. Since all elements in the first column are 4, we can factor out 4. The determinant then has two identical columns (C1 and C3 are both 1s after factoring out 4), so the value of the determinant is 0.
Correct Answer: 4