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If $a \neq b \neq c$, then one value of $x$ which satisfies the equation $\left|\begin{array}{ccc}0 & \mathrm{x}-\mathrm{a} & \mathrm{x}-\mathrm{b} \\ \mathrm{x}+\mathrm{a} & 0 & \mathrm{x}-\mathrm{c} \\ \mathrm{x}+\mathrm{b} & \mathrm{x}+\mathrm{c} & 0\end{array}\right|=0$ is given by
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$\left|\begin{array}{ccc}0 & x-a & x-b \\ x+a & 0 & x-c \\ x+b & x+c & 0\end{array}\right|=0$
We know, the value of symmetric matrix's determinant is $0\left|\begin{array}{ccc}0 & -a & -b \\ a & 0 & -c \\ b & c & 0\end{array}\right|=0$
In the given matrix, determinant is 0, if $x=0$
We know, the value of symmetric matrix's determinant is $0\left|\begin{array}{ccc}0 & -a & -b \\ a & 0 & -c \\ b & c & 0\end{array}\right|=0$
In the given matrix, determinant is 0, if $x=0$
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