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JEE Main · 2021 · 26 Aug 2021 · Shift 2 · Matrices

Let A be a

3 \times 3

real matrix. If

det (2 adj (2 Adj (adj (2 A))) = 2^{41}

, then the value of

det (A^{2})

equals ______.

Answer: 4

Solution

For a

3 \times 3

matrix

M

,

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

and

det (2 M) = 2^{3} det M

. Writing

d = det A

and working outwards,

det (2 A) = 8 d, det (adj (2 A)) = (8 d)^{2} = 2^{6} d^{2}

det (2 adj (adj (2 A))) = 2^{3} (2^{6} d^{2})^{2} = 2^{15} d^{4}

det (2 adj (2 adj (adj (2 A)))) = 2^{3} (2^{15} d^{4})^{2} = 2^{33} d^{8}

Equating this to

2^{41}

gives

d^{8} = 2^{8}

, so

d^{2} = 4

and

det (A^{2}) = (det A)^{2} = 4

.

JEE Main · 2021 · 26 Aug 2021 · Shift 2 · Integer answer

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