Solve the following equations
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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