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Question Number 15194 by arnabpapu550@gmail.com last updated on 08/Jun/17

$$\mathrm{The}\mathrm{equation}\left|\begin{array}{ccc}x-a& x-b& x-c\\ x-b& x-c& x-a\\ x-c& x-a& x-b\end{array}\right|=0,$$$$\mathrm{where}a,b,c\mathrm{are}\mathrm{different},\mathrm{is}\mathrm{satisfied}\mathrm{by}$$

Commented by prakash jain last updated on 08/Jun/17

$$\left|\begin{array}{ccc}x-a& x-b& x-c\\ x-b& x-c& x-a\\ x-c& x-a& x-b\end{array}\right|=0,$$$$\mathrm{R}1=\mathrm{R}1+\mathrm{R}2+\mathrm{R}3$$$$\left|\begin{array}{ccc}3x-a-b-c& 3x-a-b-c& 3x-c-b-a\\ x-b& x-c& x-a\\ x-c& x-a& x-b\end{array}\right|=0,$$$$(3x-a-b-c)\left|\begin{array}{ccc}1& 1& 1\\ x-b& x-c& x-a\\ x-c& x-a& x-b\end{array}\right|=0,$$$$\mathrm{C}1=\mathrm{C}1-\mathrm{C}2,\mathrm{C}2=\mathrm{C}2-\mathrm{C}3$$$$(3x-a-b-c)\left|\begin{array}{ccc}0& 0& 1\\ c-b& a-c& x-a\\ a-c& b-a& x-b\end{array}\right|=0,$$$$3x-a-b-c=0\Rightarrow x=\frac{a+b+c}{3}$$

Commented by arnabpapu550@gmail.com last updated on 08/Jun/17

$$\mathrm{excellent}!$$$$$$

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