Let A be a non-singular matrix of order 3. If $\det\big(3\,\operatorname{adj}(2\,\operatorname{adj}((\det A)A))\big)=3^{-13}\cdot2^{-10}$ and $\det\big(3\,\operatorname{adj}(2A)\big)=2^{m}\cdot3^{n}$, then $|3m+2n|$ is equal to

JEE Main · 2024 · 09 Apr 2024 · Shift 1 · Matrices

Let A be a non-singular matrix of order 3. If

det (3 adj (2 adj ((det A) A))) = 3^{- 13} \cdot 2^{- 10}

and

det (3 adj (2 A)) = 2^{m} \cdot 3^{n}

, then

∣3 m + 2 n ∣

is equal to

Answer: 14

For order 3,

adj (c M) = c^{2} adj (M)

det (adj M) = (det M)^{2}

, and

adj (adj A) = (det A) A

. Let

d = det A

. The inner expression simplifies to

3 adj (2 adj (d A)) = 12 d^{5} A

, whose determinant is

1 2^{3} d^{16} = 2^{6} 3^{3} d^{16} = 2^{- 10} 3^{- 13}

, giving

d = 2^{- 1} 3^{- 1}

. Then

det (3 adj (2 A)) = det (12 adj A) = 1 2^{3} (det A)^{2} = 2^{6} 3^{3} \cdot 2^{- 2} 3^{- 2} = 2^{4} 3^{1}

, so

m = 4, n = 1

and

∣3 m + 2 n ∣ = ∣12 + 2∣ = 14

JEE Main · 2024 · 09 Apr 2024 · Shift 1 · Integer answer