Matrices & DeterminantshardFree

Matrices & Determinants: Vmatrix Vmatrix Vmatrix Vmatrix Equals

JEE Maths question with a full step-by-step solution.

Question
If Δ2=∣y5z6(z3−y3)x4z6(x3−z3)y4z5(y3−z3)y5z3(y6−z6)xz3(z6−x6)xy2(x6−y6)y5z3(z3−y3)xz3(x3−z3)xy2(y3−x3)∣\Delta_2 = \begin{vmatrix} y^5z^6(z^3-y^3) & x^4z^6(x^3-z^3) & y^4z^5(y^3-z^3) \\ y^5z^3(y^6-z^6) & xz^3(z^6-x^6) & xy^2(x^6-y^6) \\ y^5z^3(z^3-y^3) & xz^3(x^3-z^3) & xy^2(y^3-x^3) \end{vmatrix} and Δ1=∣xyz3x4y5z6x7y8z9∣\Delta_1 = \begin{vmatrix} x & y & z^3 \\ x^4 & y^5 & z^6 \\ x^7 & y^8 & z^9 \end{vmatrix}, then Δ1Δ2\Delta_1\Delta_2 equals:
AΔ13\Delta_1^3correct
BΔ22\Delta_2^2
CΔ14\Delta_1^4
DNone of these
Solution
Step 1: Identify the structure of Δ2\Delta_2 The matrix Δ2\Delta_2 is formed by replacing each entry of Δ1\Delta_1 with its corresponding cofactor. By the standard result, the determinant of the cofactor matrix equals (det⁡Δ1)n−1(\det \Delta_1)^{n-1} for an n×nn \times n matrix. For n=3n = 3:
Δ2=Δ12\Delta_2 = \Delta_1^2
 Step 2: Compute Δ1Δ2\Delta_1\Delta_2
Δ1Δ2=Δ1⋅Δ12=Δ13\Delta_1\Delta_2 = \Delta_1 \cdot \Delta_1^2 = \Delta_1^3
 Answer: (1)
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