Let A be a matrix of order $3 \times 3$ and $\operatorname{det}(\mathrm{A})=2$. Then $\operatorname{det}\left(\operatorname{det}(\mathrm{A}) \operatorname{adj}\left(5 \operatorname{adj}\left(\mathrm{~A}^{3}\right)\right)\right)$ is equal to

JEE Main · 2022 · 28 Jun 2022 · Shift 1 · Matrices

Let A be a matrix of order

3 \times 3

and

det (A) = 2

. Then

det (det (A) adj (5 adj (A^{3})))

is equal to

Answer: (A)

Using properties of determinants and adjoints:

Let

∣ A ∣ = 2

. Then:

det (det (A) adj (5 adj (A^{3}))) = det (2 adj (5 adj (A^{3})))

= 2^{3} ∣ adj (5 adj (A^{3})) ∣

= 8 \cdot ∣5 adj (A^{3}) ∣^{2}

= 8 \cdot 5^{6} \cdot ∣ adj (A^{3}) ∣^{2}

= 8 \cdot 5^{6} \cdot (∣ A^{3} ∣^{2})^{2}

= 8 \cdot 5^{6} \cdot 2^{12} = 2^{15} \cdot 5^{6} = 2^{9} \cdot 1 0^{6} = 512 \times 1 0^{6}

JEE Main · 2022 · 28 Jun 2022 · Shift 1 · Single correct option