Evaluate 2&3&7 13&17&5 15&20&12 ^2 - Vidyadip

Question

Evaluate $\begin{vmatrix}2&3&7\\13&17&5\\15&20&12\end{vmatrix}^2$

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Answer

Let $\text{A}=\begin{vmatrix}2&3&7\\13&17&5\\15&20&12\end{vmatrix}$
$\Rightarrow\text{A}=2(204-100)-3(156-75)+7(260-255)$
$\Rightarrow\text{A}=2(104)-3(81)+7(5)$
$\Rightarrow\text{A}=208-243+35$
$\Rightarrow\text{A}=243-243=0$
$\because\begin{vmatrix}2&3&7\\13&17&5\\15&20&12\end{vmatrix}=0$
$\Rightarrow\begin{vmatrix}2&3&7\\13&17&5\\15&20&12\end{vmatrix}^2=0^2=0$ $\big[\therefore\text{det } \text{A}^2 = (\text{det} \text{A})^2\big]$

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