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$\left|\begin{array}{ccc}1 & 1 & 1 \\ a & b & c \\ a^{2}-b c & b^{2}-c a & c^{2}-a b\end{array}\right|$ is equal to
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Let $\Delta=\left|\begin{array}{ccc}1 & 1 & 1 \\ a & b & c \\ a^{2}-b c & b^{2}-c a & c^{2}-a b\end{array}\right|$
On applying $C_{3} \rightarrow C_{3}-C_{2}$ and $C_{2} \rightarrow C_{2}-C_{1}$, we get
$$
\begin{aligned}
\Delta=&\left|\begin{array}{ccc}
1 & 0 & 0 \\
a & b-a & c-b \\
a^{2}-b c & +c(b-a) & +a(c-b)
\end{array}\right| \\
=& 1[(b-a)(c-b)(c+b+a)\\
&-(c-b)(b-a)(b+a+c)]=0
\end{aligned}
$$
On applying $C_{3} \rightarrow C_{3}-C_{2}$ and $C_{2} \rightarrow C_{2}-C_{1}$, we get
$$
\begin{aligned}
\Delta=&\left|\begin{array}{ccc}
1 & 0 & 0 \\
a & b-a & c-b \\
a^{2}-b c & +c(b-a) & +a(c-b)
\end{array}\right| \\
=& 1[(b-a)(c-b)(c+b+a)\\
&-(c-b)(b-a)(b+a+c)]=0
\end{aligned}
$$
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