If $ \mathrm{A} $ is an invertible matrix of order $ 3 $ and $ \mathrm{B} $ is another matrix of the same order as of $ \mathrm{A} $, such that $ |B|=2, A^{T}|A| B=A|B| B^{T} . $ If $ \left|A B^{-1} \operatorname{adj}\left(A^{T} B\right)^{-1}\right|=K $, then the value of $ 4 K $ is equal to

Question

If $ \mathrm{A} $ is an invertible matrix of order $ 3 $ and $ \mathrm{B} $ is another matrix of the same order as of $ \mathrm{A} $, such that $ |B|=2, A^{T}|A| B=A|B| B^{T} . $ If $ \left|A B^{-1} \operatorname{adj}\left(A^{T} B\right)^{-1}\right|=K $, then the value of $ 4 K $ is equal to,

MARKS App · Accepted Answer

∵ A T A B = A B B T Taking determinant on both sides, we get, A T A B = A B B T ⇒ A = 2 Now, A B - 1 a d j A T B - 1 | A | × 1 | B | × 1 | a d j ( A T B ) | A × 1 B × 1 A T B 2 = A B 3 A 2 = 1 16 i.e. 4 K = 1 4 = 0 . 25

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