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Using the property of determinants and without expanding, prove that
$\left|\begin{array}{lll}x & a & x+a \\ y & b & y+b \\ z & c & z+c\end{array}\right|=0$
$\left|\begin{array}{lll}x & a & x+a \\ y & b & y+b \\ z & c & z+c\end{array}\right|=0$
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L.H.S. $=\left|\begin{array}{lll}x & a & x \\ y & b & y \\ z & c & z\end{array}\right|+\left|\begin{array}{lll}x & a & a \\ y & b & b \\ z & c & c\end{array}\right|\left(\mathrm{C}_1=\mathrm{C}_3\right.$ and $\left.\mathrm{C}_2=\mathrm{C}_3\right)$ $=0+0=0=\mathrm{R}^2 \mathrm{~S}$ S $=0+0=0=$ R.H.S.
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