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Matrices & Determinants: Matrix Adj Adj Equals

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

Question
If AA is a 3×33 \times 3 matrix and det⁡A=5\det A = 5, then det⁡(Adj(Adj A))\det\big(\text{Adj}(\text{Adj}\,A)\big) equals:
A525^2
B535^3
C545^4correct
D555^5
Solution
Step 1: Apply the iterated adjoint identity For an n×nn \times n matrix, det⁡(Adj(Adj A))=∣A∣(n−1)2\det\big(\text{Adj}(\text{Adj}\,A)\big) = |A|^{(n-1)^2}. Step 2: Substitute n=3n = 3
(n−1)2=4  ⟹  det⁡(Adj(Adj A))=54(n-1)^2 = 4 \implies \det\big(\text{Adj}(\text{Adj}\,A)\big) = 5^4
 Answer: (3)
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