If $\left|\begin{array}{rrr}-a^2 & a b & a c \\ a b & -b^2 & b c \\ a c & b c & -c^2\end{array}\right|=\lambda a^2 b^2 c^2$, then the value of $\lambda$ is :

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$\left|\begin{array}{rrr}-a^2 & a b & a c \\ a b & -b^2 & b c \\ a c & b c & -c^2\end{array}\right|=\lambda a^2 b^2 c^2$

taking factor a, b,c from R₁, R₂, R₃ respectively (rows of matrix)

so  $a b c\left|\begin{array}{ccc}-a & b & c \\ a & -b & c \\ a & b & -c\end{array}\right|=\lambda a^2 b^2 c^2$

taking out factor a, b, c from C₁, C₂, C₃ respectively (Columns of matrix)

$\Rightarrow a^2 b^2 c^2\left|\begin{array}{ccc}-1 & 1 & 1 \\ 1 & -1 & 1 \\ 1 & 1 & -1\end{array}\right|=\lambda a^2 b^2 c^2$

$\Rightarrow \lambda = \left|\begin{array}{ccc}-1 & 1 & 1 \\ 1 & -1 & 1 \\ 1 & 1 & -1\end{array}\right| \Rightarrow \lambda=\left|\begin{array}{rrr}
-1 & 0 & 0 \\
1 & 0 & 2 \\
1 & 2 & 0
\end{array}\right| $

using operations 

C₂ → C₁ + C₂

C₃ → C₁ + C₃

calculating determinant across R₁ row

$\lambda=-1\left|\begin{array}{ll}0 & 2 \\ 2 & 0\end{array}\right|+0+0=-1(-4)$

$\lambda = 4$