Let $$A=\left[\begin{array}{ccc}2 & 1 & 0 \\ 1 & 2 & -1 \\ 0 & -1 & 2\end{array}\right]$$. If $|\operatorname{adj}(\operatorname{adj}(\operatorname{adj} 2 A))|=(16)^{n}$, then $n$ is equal to :
Solution
We have,
<br/><br/>$$
\begin{aligned}
& |\mathrm{A}|=\left|\begin{array}{ccc}
2 & 1 & 0 \\
1 & 2 & -1 \\
0 & -1 & 2
\end{array}\right|=2(4-1)-1(2-0)+0 \\\\
& =6-2=4 \\\\
& \text { So, }|2 \mathrm{~A}|=2^3|\mathrm{~A}|=8 \times 4=32 \\\\
& \text { Now, }|\operatorname{adj}(\operatorname{adj}(\operatorname{adj} 2 \mathrm{~A}))|=|2 \mathrm{~A}|^{(n-1)^3} \\\\
& =(32)^{2^3}=32^8 \\\\
& \Rightarrow 16^n=(32)^8=2^8 \times 16^8 \\\\
& \Rightarrow 16^n=16^{2+8} \Rightarrow n=10
\end{aligned}
$$
<br/><br/><b>Concepts :</b>
<br/><br/>(a) $|k \mathrm{~A}|=k^n|\mathrm{~A}|$
<br/><br/>(b) $|\operatorname{adj} \mathrm{A}|=|\mathrm{A}|^{n-1}$
About this question
Subject: Mathematics · Chapter: Matrices and Determinants · Topic: Types of Matrices and Operations
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