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Matrices & Determinants: Non Singular Matrices Order Adj Equals

JEE Maths question with a full step-by-step solution.

Question
If AA and BB are two non-singular matrices of order 3 such that AAT=2IAA^T = 2I and A−1=AT−A adj(2B−1)A^{-1} = A^T - A\,\text{adj}(2B^{-1}), then det⁡(B)\det(B) equals:
A44
B424\sqrt{2}
C1616
D16216\sqrt{2}correct
Solution
Step 1: Take the determinant of AAT=2IAA^T = 2I
∣A∣2=∣2I∣=23=8|A|^2 = |2I| = 2^3 = 8
 Step 2: Multiply the given relation on the left by AA
AA−1=AAT−A2 adj(2B−1)  ⟹  I=2I−A2 adj(2B−1)A A^{-1} = AA^T - A^2\,\text{adj}(2B^{-1}) \implies I = 2I - A^2\,\text{adj}(2B^{-1})
A2 adj(2B−1)=IA^2\,\text{adj}(2B^{-1}) = I
 Step 3: Take the determinant
∣A∣2 ∣adj(2B−1)∣=1  ⟹  8 ∣adj(2B−1)∣=1|A|^2\,|\text{adj}(2B^{-1})| = 1 \implies 8\,|\text{adj}(2B^{-1})| = 1
 Using ∣adj(M)∣=∣M∣2|\text{adj}(M)| = |M|^2 for order 3 and ∣2B−1∣=23∣B∣|2B^{-1}| = \dfrac{2^3}{|B|}:
8⋅(8∣B∣)2=1  ⟹  512∣B∣2=1  ⟹  ∣B∣2=512  ⟹  ∣B∣=1628\cdot\left(\frac{8}{|B|}\right)^2 = 1 \implies \frac{512}{|B|^2} = 1 \implies |B|^2 = 512 \implies |B| = 16\sqrt{2}
 Answer: (4)
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