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Mathematics Question on Determinants

By using properties of determinants, show that: ∣−a2abac\ba−b2bc\cacb−c2∣\begin{vmatrix}-a^2&ab&ac\\\ba&-b^2&bc\\\ca&cb&-c^2\end{vmatrix}=4a2b2c2

Answer

△=∣−a2abac\ba−b2bc\cacb−c2∣\begin{vmatrix}-a^2&ab&ac\\\ba&-b^2&bc\\\ca&cb&-c^2\end{vmatrix}

=abc∣−abc\a−bc\ab−c∣abc\begin{vmatrix}-a&b&c\\\a&-b&c\\\a&b&-c\end{vmatrix} [Taking out factors a,b,c from R1,R2and R3]

=a2b2c2 ∣−111\1−11\11−1∣\begin{vmatrix}-1&1&1\\\1&-1&1\\\1&1&-1\end{vmatrix} [Taking out factors a,b,c from C1,C2and C3]
Applying R2 → R2 + R1 and R3 → R3 + R1, we have:

△=a2b2c2∣−111\002\020∣\begin{vmatrix}-1&1&1\\\0&0&2\\\0&2&0\end{vmatrix}
=a2b2c2(-1)∣02\20∣\begin{vmatrix}0&2\\\2&0\end{vmatrix}
=-a2b2c2(0-4)

=4a2b2c2