The value of the determinant $\left|\begin{array}{lll}1 & a & a^{2}-b c \ 1 & b & b^{2}-a c \ 1 &

Question

The value of the determinant $\left|\begin{array}{lll}1 & a & a^{2}-b c \ 1 & b & b^{2}-a c \ 1 & c & c^{2}-a b\end{array}ight|$ is .......................

Professor PC Thomas Classes · Answer

Given determinant
$$
\begin{aligned}
& =\left|\begin{array}{lll}
1 & a & a^2 \
1 & b & b^2 \
1 & c & c^2
\end{array}ight|-\left|\begin{array}{lll}
1 & a & b c \
1 & b & a c \
1 & c & a b
\end{array}ight| \
& =\left|\begin{array}{lll}
1 & a & a^2 \
1 & b & b^2 \
1 & c & c^2
\end{array}ight|-\frac{1}{a b c}\left|\begin{array}{lll}
a & a^2 & a b c \
b & b^2 & a b c \
c & c^2 & a b c
\end{array}ight| \
& =\left|\begin{array}{lll}
1 & a & a^2 \
1 & b & b^2 \
1 & c & c^2
\end{array}ight|-\left|\begin{array}{lll}
1 & a & a^2 \
1 & b & b^2 \
1 & c & c^2
\end{array}ight|=0
\end{aligned}
$$