If A is a square matrix of order 3 such that det(A)=3 and det(adj(-4 adj(-3 adj(3 adj((2 A)⁻¹)))))=2ᵐ 3ⁿ, then m+2 n is equal to :

Solution

Maths sir · 4 steps

Peel the brackets from the inside out and track only the determinant as powers of 2 and 3.

1. Two rules. For any matrix B of order 3, a scalar comes out cubed: det(kB) = k³ det B. The adjoint squares the determinant: det(adj B) = (det B)². Also det(B⁻¹) = 1/(det B).
2. Start inside. det(2A) = 2³ × 3 = 24. Keep every determinant as a power of 2 times a power of 3, so each step only changes the exponents.

   Layer	Rule used	Determinant
   (2A)⁻¹	reciprocal	2⁻³ × 3⁻¹
   adj((2A)⁻¹)	square	2⁻⁶ × 3⁻²
   3 adj(…)	multiply by 3³	2⁻⁶ × 3¹
   adj(3 adj(…))	square	2⁻¹² × 3²
   -3 adj(…)	multiply by (-3)³ = -3³	-2⁻¹² × 3⁵
   adj(-3 adj(…))	square	2⁻²⁴ × 3¹⁰
   -4 adj(…)	multiply by (-4)³ = -2⁶	-2⁻¹⁸ × 3¹⁰
   adj(-4 adj(…))	square	2⁻³⁶ × 3²⁰

3. Read off exponents. 2ᵐ 3ⁿ = 2⁻³⁶ × 3²⁰, so m = -36 and n = 20. The minus signs vanish because the last step squares the determinant.
4. So. m + 2n = -36 + 40 = 4, which is option (b).

The idea to keep. For a matrix of order p, a scalar k comes out as kᵖ and adj raises the determinant to the power p-1. Write everything as prime powers and update the exponents layer by layer.

Answer from the NTA final answer key. Solution by PrepWiser.