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If $A$ is a matrix of order $4 \operatorname{such}$ that $A(\operatorname{adj} A)$ $=10 \mathrm{I}$, then $|\operatorname{adj} A|$ is equal to
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1000
Given, $A(\operatorname{adj} A)=10 I$
We know that $A(\operatorname{adj} A)=|A| I$
$\begin{array}{ll}
\therefore & 10 I=|A| I \\
\Rightarrow & |A|=10
\end{array}$
We know that $|\operatorname{adj} A|=|A|^{n-1}$, where $n$ is order of $A$
$\therefore|\operatorname{adj} A|=|A|^{4-1}=10^3=1000$
We know that $A(\operatorname{adj} A)=|A| I$
$\begin{array}{ll}
\therefore & 10 I=|A| I \\
\Rightarrow & |A|=10
\end{array}$
We know that $|\operatorname{adj} A|=|A|^{n-1}$, where $n$ is order of $A$
$\therefore|\operatorname{adj} A|=|A|^{4-1}=10^3=1000$
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