$$
\begin{aligned}
&\\
&\text { LHS }=\left|\begin{array}{ccc}
x a & y b & z c \\
y c & z a & x b \\
z b & x c & y a
\end{array}\right|\\
&=x a(z a \cdot y a-x b \cdot x c)-y b(y c \cdot y a-x b \cdot z b)+\\
&=x y z\left(a^3+b^3+c^3\right)-a b c\left(x^3+y^3+z^3\right) \quad z c(y c \cdot x c-z a \cdot z b)\\
&=x y z\left(a^3+b^3+c^3\right)-a b c\left(x^3+y^3+z^3\right)\\
&=x y z\left(a^3+b^3+c^3\right)-a b c(3 x y z)\\
&\left[\because x+y+z=0=x^3+y^3+z^3-3 x y z\right]\\
&=x y z\left(a^3+b^3+c^3-3 a b c\right)\\
&\text { Now, RHS }=x y z\left|\begin{array}{ccc}
a & b & c \\
c & a & b \\
b & c & a
\end{array}\right|\\
&=x y z\left|\begin{array}{lll}
a+b+c & b & c \\
a+b+c & a & b \\
a+b+c & c & a
\end{array}\right|
\end{aligned}
$$

$$
\begin{aligned}
&\left[\because \mathrm{C}_1 \rightarrow \mathrm{C}_1+\mathrm{C}_2+\mathrm{C}_3\right]\\
&=x y z(a+b+c)\left|\begin{array}{lll}
1 & b & c \\
1 & a & b \\
1 & c & a
\end{array}\right|\\
&=x y z(a+b+c)\left|\begin{array}{ccc}
0 & b-c & c-a \\
0 & a-c & b-a \\
1 & c & a
\end{array}\right|\\
&\left[\because \mathrm{R}_1 \rightarrow \mathrm{R}_1-\mathrm{R}_3 \text { and } \mathrm{R}_2 \rightarrow \mathrm{R}_2-\mathrm{R}_3\right]\\
&=x y z(a+b+c)[1(b-c)(b-a)-(a-c)(c-a)]\\
&=x y z(a+b+c)\left(a^2+b^2+c^2-a b-b c-c a\right)\\
&=x y z\left(a^3+b^3+c^3-3 a b c\right)\\
&\text { From Eqs. (i) and(ii), }\\
&\text { LHS }=\text { RHS }
\end{aligned}
$$