If A and B are two non-singular matrices of order 3 such that AAT=2I and A−1=AT−A . adj 2B−1 , then det⁡ (B) equals

Given that AAT=2I⇒|A|2=8 And A−1=AT−A adj⁡2B−1⇒I=AAT−A2adj⁡2B−1⇒I=2I−A2adj⁡2B−1⇒A2 adj 2B−1=I⇒A2∥adj⁡2B−1=1⇒82B−12=1        ∵ adj A=An-1⇒8×64|B|2=1         ∵ B-1=1Band kA=k3A=23B-12⇒|B|2=64×8∴|B|=±162