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JEE Main · 2023 · 31 Jan 2023 · Shift 2 · Matrices

Let A be a

n \times n

matrix such that

∣ A ∣ = 2

. If the determinant of the matrix

Adj (2. Adj (2 A^{- 1}))

is

2^{84}

, then n is equal to ______.

Answer: 5

Solution

Using

∣ k B ∣ = k^{n} ∣ B ∣

and

∣ Adj B ∣ = ∣ B ∣^{n - 1}

:

∣2 A^{- 1} ∣ = \frac{2 ^{n}}{2} = 2^{n - 1}

, so

∣ Adj (2 A^{- 1}) ∣ = 2^{(n - 1)^{2}}

and

∣2 Adj (2 A^{- 1}) ∣ = 2^{n + (n - 1)^{2}} .

Hence

Adj (2 Adj (2 A^{- 1})) = 2^{(n - 1) [n + (n - 1)^{2}]} = 2^{(n - 1) (n^{2} - n + 1)} .

Setting

(n - 1) (n^{2} - n + 1) = 84

, the value

4 \cdot 21 = 84

occurs at

n = 5

, so

n = 5.

JEE Main · 2023 · 31 Jan 2023 · Shift 2 · Integer answer

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