Determinants and Matrices — Maths STD 11 Science — Question

$
\begin{gathered}
=\frac{1}{a}\left|\begin{array}{lll}
a & a^2 & a b c \\
1 & b & c a \\
1 & c & a b
\end{array}\right| \\
\mathrm{R}_2 \rightarrow \mathrm{bR}_2
\end{gathered}
$
$=\frac{1}{a} \times \frac{1}{b}\left|\begin{array}{lll}a & a^2 & a b c \\ b & b^2 & a b c \\ 1 & c & a b\end{array}\right|$
$
\mathrm{R}_3 \rightarrow \mathrm{cR}_3
$
$=\frac{1}{a} \times \frac{1}{b} \times \frac{1}{c}\left|\begin{array}{lll}a & a^2 & a b c \\ b & b^2 & a b c \\ c & c^2 & a b c\end{array}\right|$
$=\frac{1}{a b c} \times a b c\left|\begin{array}{lll}a & a^2 & 1 \\ b & b^2 & 1 \\ c & c^2 & 1\end{array}\right|$
(taking abc common from $\mathrm{C}_3$ )
$
\begin{aligned}
& =\left|\begin{array}{lll}
a & a^2 & 1 \\
b & b^2 & 1 \\
c & c^2 & 1
\end{array}\right| \\
& \mathrm{C}_1 \leftrightarrow \mathrm{C}_3 \\
& =(-1)\left|\begin{array}{lll}
1 & a^2 & a \\
1 & b^2 & b \\
1 & c^2 & c
\end{array}\right| \\
& \mathrm{C}_2 \leftrightarrow \mathrm{C}_3 \\
&
\end{aligned}
$
$
\begin{gathered}
=(-1)\left|\begin{array}{lll}
1 & a^2 & a \\
1 & b^2 & b \\
1 & c^2 & c
\end{array}\right| \\
\mathrm{C}_2 \leftrightarrow \mathrm{C}_3
\end{gathered}
$
$=(-1)(-1)\left|\begin{array}{lll}1 & a & a^2 \\ 1 & b & b^2 \\ 1 & c & c^2\end{array}\right|$
$
=\left|\begin{array}{lll}
1 & a & a^2 \\
1 & b & b^2 \\
1 & c & c^2
\end{array}\right|=\text { R.H.S. }
$