If A is a $3 \times 3$ matrix and $|\mathrm{A}|=2$, then $\left|3 a d j\left(|3 A| A^{2}\right)\right|$ is equal to

JEE Main · 2023 · 10 Apr 2023 · Shift 1 · Matrices

If A is a

3 \times 3

matrix and

∣ A ∣ = 2

, then

3 a d j (∣3 A ∣ A^{2})

is equal to

Answer: (A)

For a

3 \times 3

matrix,

∣ k M ∣ = k^{3} ∣ M ∣

and

∣ adj M ∣ = ∣ M ∣^{2}

. Here

∣3 A ∣ = 3^{3} ∣ A ∣ = 27 \cdot 2 = 54

is a scalar, so the matrix inside the adjugate is

54 A^{2}

, with

∣54 A^{2} ∣ = 5 4^{3} ∣ A ∣^{2} = 5 4^{3} \cdot 4

. Hence

∣ adj (54 A^{2}) ∣ = (5 4^{3} \cdot 4)^{2} = 5 4^{6} \cdot 16

, and

3 adj (54 A^{2}) = 3^{3} \cdot 5 4^{6} \cdot 16

. Writing

54 = 2 \cdot 3^{3}

gives

5 4^{6} = 2^{6} \cdot 3^{18}

, so the value is

3^{3} \cdot 2^{6} \cdot 3^{18} \cdot 2^{4} = 3^{21} \cdot 2^{10} = 3^{11} \cdot 6^{10}

JEE Main · 2023 · 10 Apr 2023 · Shift 1 · Single correct option