JEE Main 2021 · 26 August · evening · Numerical answer · +4 / −0
Let A be a 3 × 3 real matrix. If det(2Adj(2 Adj(Adj(2A)))) = 2⁴¹, then the value of det(A²) equal __________.
Answer and solution
Answer 4
Solution
Maths sir · 2 steps
Peel this from the inside out, using just two rules, and track only powers of 2 and |A|.
- Two rules. For a 3 × 3 matrix M, det(kM) = k³ det M and det(Adj M) = (det M)². Let det A = d and apply these one layer at a time.
Matrix Rule used Determinant 2A k³ with k = 2 8d = 2³d Adj(2A) square it 2⁶d² Adj(Adj(2A)) square it 2¹²d⁴ 2 Adj(Adj(2A)) multiply by 2³ 2¹⁵d⁴ Adj(2 Adj(Adj(2A))) square it 2³⁰d⁸ 2 Adj(2 Adj(Adj(2A))) multiply by 2³ 2³³d⁸ 2³³d⁸ = 2⁴¹ ⇒ d⁸ = 2⁸ ⇒ d² = 4
- So. det(A²) = (det A)² = d² = 4. It does not matter whether d = 2 or d = -2.
The common slip: 2. Stopping at det A = 2 gives 2, but the question asks for det(A²) = (det A)² = 4.
The idea to keep. For an n × n matrix, a scalar k outside pulls out as kⁿ, and Adj raises the determinant to the power n-1. Nested questions are just these two rules applied layer by layer.
Answer from the NTA final answer key. Solution by PrepWiser.