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If A be a square matrix of order 3 such that ∣A∣=2|A| = 2, then ∣adj(2A)∣|adj(2A)| is equal to

Solution

✅ Correct Option: 4

Given that AA is a square matrix of order 3 and ∣A∣=2|A| = 2.


For a matrix of order nn, when multiplying by a constant kk:

∣kA∣=kn×∣A∣|kA| = k^n \times |A|

Here, k=2k = 2 and n=3n = 3:

∣2A∣=23×∣A∣|2A| = 2^3 \times |A|

∣2A∣=8×2|2A| = 8 \times 2

∣2A∣=16|2A| = 16


For any square matrix BB of order nn:

∣adj(B)∣=∣B∣n−1|adj(B)| = |B|^{n-1}

Let B=2AB = 2A, where ∣2A∣=16|2A| = 16 and n=3n = 3:

∣adj(2A)∣=∣2A∣3−1|adj(2A)| = |2A|^{3-1}

∣adj(2A)∣=∣2A∣2|adj(2A)| = |2A|^2

∣adj(2A)∣=(16)2|adj(2A)| = (16)^2

∣adj(2A)∣=256|adj(2A)| = 256

Therefore, ∣adj(2A)∣=256|adj(2A)| = 256.

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