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\newcommand{\un}{\underline}
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{\bf Question}

Prove the rule that a determinant vanishes when two rows are
identical.


\medskip

{\bf Answer}

Let my determinant be $\bigtriangleup$ where

$\bigtriangleup=\left|\begin{array}{ccc} a_1 & b_1 & c_1\\ a_1 &
b_1 & c_1\\ a_3 & b_3 & c_3 \end{array} \right|$ by rule that
interchanging 2 rows means $\bigtriangleup \rightarrow
-\bigtriangleup$ and interchanging first 2 rows

$\ =-\left|\begin{array}{ccc} a_1 & b_1 & c_1\\ a_1 & b_1 & c_1\\
a_3 & b_3 & c_3 \end{array} \right|=-\bigtriangleup$

Thus

$\bigtriangleup=-\bigtriangleup$

$\Rightarrow 2\bigtriangleup=0$

$\Rightarrow \un{\bigtriangleup=0}$

Hence 2 rows identical $\Rightarrow \bigtriangleup=0$.

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