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Question Number 21174 by akinyeni last updated on 15/Sep/17

$$\mathrm{The}\mathrm{factors}\mathrm{of}\left|\begin{array}{ccc}x& a& b\\ a& x& b\\ a& b& x\end{array}\right|\mathrm{are}$$

Answered by $@ty@m last updated on 15/Sep/17

$$=\left|\begin{array}{ccc}x& a& b\\ 0& x-b& b-x\\ a& b& x\end{array}\right|,by{R}_2\to R_2-R_3$$$$=(x-b)\left|\begin{array}{ccc}x& a& b\\ 0& 1& -1\\ a& b& x\end{array}\right|$$$$=(x-b)\left|\begin{array}{ccc}x& a+b& b\\ 0& 0& -1\\ a& b+x& x\end{array}\right|,by{C}_2\to C_2+C_3$$$$=(x-b)\left|\begin{array}{cc}x& a+b\\ a& b+x\end{array}\right|$$$$=(x-b)\left|\begin{array}{cc}x-a& a-x\\ a& b+x\end{array}\right|,by{R}_1\to R_1-R_2$$$$=(x-b)(x-a)\left|\begin{array}{cc}1& -1\\ a& b+x\end{array}\right|$$$$=(x-a)(x-b)\left|\begin{array}{cc}0& -1\\ a+b+x& b+x\end{array}\right|,by{C}_1\to C_1+C_2$$$$=(x-a)(x-b)(a+b+x)$$$$$$

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