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If the adjoint of a $3 \times 3$ matrix $P$ is $\left[\begin{array}{lll}1 & 4 & 4 \\ 2 & 1 & 7 \\ 1 & 1 & 3\end{array}\right]$, then the possible value(s) of the determinant of $P$ is (are)
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Verified Answer
The correct answers are:
$-2$, 2
We know for a third order matrix $P$,
$|\operatorname{Adj} P|=|P|^{2}$
Where
$\begin{aligned}
&|\operatorname{Adj} P|=1(3-7)-4(6-7)+4(2-1)=4 \\
\therefore \quad &|P|^{2}=4 \mathrm{P}|P|=2 \text { or }-2
\end{aligned}$
$|\operatorname{Adj} P|=|P|^{2}$
Where
$\begin{aligned}
&|\operatorname{Adj} P|=1(3-7)-4(6-7)+4(2-1)=4 \\
\therefore \quad &|P|^{2}=4 \mathrm{P}|P|=2 \text { or }-2
\end{aligned}$
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