Question

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

Answer 1

The correct answer is 5.
∣∣​Adj(2Adj(2A−1))∣∣​
=∣∣​2Adj(Adj(2A−1))∣∣​n−1
=2n(n−1)∣∣​Adj(2A−1)∣∣​n−1
=2n(n−1)∣∣​(2A−1)∣∣​(n−1)(n−1)
=2n(n−1)2n(n−1)(n−1)∣∣​A−1∣∣​(n−1)(n−1)
=2n(n−1)+n(n−1)(n−1)∣A∣(n−1)21​
=2(n−1)22n(n−1)+n(n−1)(n−1)​
=2n(n−1)+n(n+1)2−(n−1)2
=2(n−1)(n2−n+1)
Now 2(n−1)(n2−n+1)
2(n−1)(n2−n+1)=284
So, n=5