Question

Let A be a $n \times n$ matrix such that $| A |=2$. If the determinant of the matrix $\text{Adj}\left(2 \cdot \text{Adj}\left(2 A ^{-1}\right)\right) \cdot$ is $2^{84}$, then $n$ is equal to ____

Answer 1

$ \left|\text{Adj}\left(2 \text{Adj}\left(2 A ^{-1}\right)\right)\right| $
$ =\left|2 \text{Adj}\left(\text{Adj}\left(2 A ^{-1}\right)\right)\right|^{ n -1} $
$ =2^{ n ( n -1)}\left|\text{Adj}\left(2 A ^{-1}\right)\right|^{ n -1} $
$ =2^{ n ( n -1)}\left|\left(2 A ^{-1}\right)\right|^{( n -1)( n -1)} $
$ =2^{ n ( n -1)} 2^{ n ( n -1)( n -1)}\left| A ^{-1}\right|^{( n -1)( n -1)} $
$=2^{ n ( n -1)+ n ( n -1)( n -1)} \frac{1}{| A |^{( n -1)^2}} $
$ =\frac{2^{ n ( n -1)+ n ( n -1)( n -1)}}{2^{( n -1)^2}} $
$ =2^{ n ( n -1)+ n ( n +1)^2-( n -1)^2} $
$ =2^{( n -1)\left( n ^2- n +1\right)} $
$ \text { Now } 2^{( n -1)\left( n ^2- n +1\right)} $
$ 2^{( n -1)\left( n ^2- n +1\right)}=2^{84} $
$ \text { So, } n =5$