Let α ∈ ( 0 , ∞ ) and A = 1 1 0 2 0 1 α 1 2 . If det ( adj ( 2 A − A T ) ⋅ adj ( A − 2 A T ) ) = 2 8 , then ( det ( A ) ) 2 is equal to:

Question

Let α ∈ ( 0 , ∞ ) and A = ​ 1 1 0 ​ 2 0 1 ​ α 1 2 ​ ​ . If det ( adj ( 2 A − A T ) ⋅ adj ( A − 2 A T ) ) = 2 8 , then ( det ( A ) ) 2 is equal to:

TestHub · Accepted Answer

​ ​ adj ( A − 2 A T ) ( 2 A − A T ) ​ = 28 ​ ( A − 2 A T ) ( 2 A − A T ) ​ = 24 ​ A − 2 A T ​ ​ 2 A − A T ​ = ± 16 ( A − 2 A T ) T = A T − 2 A ​ A − 2 A T ​ = ​ A T − 2 A ​ ⇒ ​ A − 2 A T ​ 2 = 16 ​ A − 2 A T ​ = ± 4 ​ 1 1 0 ​ 2 0 1 ​ α 1 2 ​ ​ − ​ 2 4 2 α ​ 2 0 2 ​ 0 2 4 ​ ​ ​ − 1 − 3 − 2 α ​ 0 0 − 1 ​ α − 1 − 2 ​ ​ 1 + 3 α = 4 3 α = 3 α = 1 ∣ A ∣ = ​ 1 1 0 ​ 2 0 1 ​ 1 1 2 ​ ​ = − 1 − 3 = − 4 ∣ A ∣ 2 = 16 ​

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