Let A be a $n \times n$ matrix such that $|\mathrm{A}|=2$. If the determinant of the matrix $\operatorname{Adj}\left(2 \cdot \operatorname{Adj}\left(2 \mathrm{~A}^{-1}\right)\right) \cdot$ is $2^{84}$, then $\mathrm{n}$ is equal to :
Answer (integer)
5
Solution
$\because\left|\operatorname{adj}\left(2 \cdot \operatorname{adj}\left(2 A^{-1}\right)\right)\right|=2^{84}$
<br/><br/>$\Rightarrow 2^{n \cdot(n-1)}\left|\operatorname{adj}\left(2 A^{-1}\right)\right|^{(n-1)}=2^{84}$
<br/><br/>$\Rightarrow 2^{n(n-1)}\left|2 A^{-1}\right|^{(n-1)^{2}}=2^{84}$
<br/><br/>$\Rightarrow 2^{n(n-1)} \cdot 2^{n(n-1)^{2}} \cdot \frac{1}{|A|^{(n-1)^{2}}}=2^{84}$
<br/><br/>$\Rightarrow 2^{n(n-1)+n(n-1)^{2}-(n-1)^{2}}=2^{84}\{\because|4|=2\}$
<br/><br/>$\therefore n(n-1)+(n-1)^{3}=84$
<br/><br/>$\therefore n=5$
About this question
Subject: Mathematics · Chapter: Matrices and Determinants · Topic: Types of Matrices and Operations
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