Question:
Evaluate $\left|\begin{array}{ccc}2 & 3 & 7 \\ 13 & 17 & 5 \\ 15 & 20 & 12\end{array}\right|^{2}$
Solution:
Let $A=\left|\begin{array}{ccc}2 & 3 & 7 \\ 13 & 17 & 5 \\ 15 & 20 & 12\end{array}\right|=2(204-100)-3(156-75)+7(260-255)$
$\Rightarrow A=2(104)-3(81)+7(5)$
$\Rightarrow A=208-243+35$
$\Rightarrow A=243-243=0$
$\therefore\left|\begin{array}{ccc}2 & 3 & 7 \\ 13 & 17 & 5 \\ 15 & 20 & 12\end{array}\right|=0$
$\Rightarrow\left|\begin{array}{ccc}2 & 3 & 7 \\ 13 & 17 & 5 \\ 15 & 20 & 12\end{array}\right|^{2}=0^{2}=0$ $\left[\because \operatorname{det} A^{2}=(\operatorname{det} A)^{2}\right]$
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