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$\left|\begin{array}{ccc}a-1 & a & b c \\ b-1 & b & c a \\ c-1 & c & a b\end{array}\right|=$
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$\left|\begin{array}{lll}a-1 & a & b c \\ b-1 & b & c a \\ c-1 & c & a b\end{array}\right|=\left|\begin{array}{lll}a & a & b c \\ b & b & c a \\ c & c & a b\end{array}\right|-\left|\begin{array}{ccc}1 & a & b c \\ 1 & b & c a \\ 1 & c & a b\end{array}\right|$
$=-\left|\begin{array}{lll}a & a^2 & 1 \\ b & b^2 & 1 \\ c & c^2 & 1\end{array}\right|=-\left|\begin{array}{ccc}a & a^2 & 1 \\ b-a & b^2-a^2 & 0 \\ c-a & c^2-a^2 & 0\end{array}\right|$ [By $\left.R_2 \rightarrow R_2-R_1 ; R_3 \rightarrow R_3-R_1\right]$
$=-(a-b)(b-c)(c-a)$
$=-\left|\begin{array}{lll}a & a^2 & 1 \\ b & b^2 & 1 \\ c & c^2 & 1\end{array}\right|=-\left|\begin{array}{ccc}a & a^2 & 1 \\ b-a & b^2-a^2 & 0 \\ c-a & c^2-a^2 & 0\end{array}\right|$ [By $\left.R_2 \rightarrow R_2-R_1 ; R_3 \rightarrow R_3-R_1\right]$
$=-(a-b)(b-c)(c-a)$
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