By using properties of determinants, show that: \(\begin{vmatrix}-a^2&ab&ac\\ba&-b^2&bc\\ca&cb&-c^2\end{vmatrix}\)=4a2b2c2
△=\(\begin{vmatrix}-a^2&ab&ac\\ba&-b^2&bc\\ca&cb&-c^2\end{vmatrix}\)
=\(abc\begin{vmatrix}-a&b&c\\a&-b&c\\a&b&-c\end{vmatrix}\) [Taking out factors a,b,c from R1,R2and R3]
=a2b2c2 \(\begin{vmatrix}-1&1&1\\1&-1&1\\1&1&-1\end{vmatrix}\) [Taking out factors a,b,c from C1,C2and C3]
Applying R2 → R2 + R1 and R3 → R3 + R1, we have:
△=a2b2c2\(\begin{vmatrix}-1&1&1\\0&0&2\\0&2&0\end{vmatrix}\)
=a2b2c2(-1)\(\begin{vmatrix}0&2\\2&0\end{vmatrix}\)
=-a2b2c2(0-4)
=4a2b2c2
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Read More: Properties of Determinants