Question

By using properties of determinants, show that: \(\begin{vmatrix}-a^2&ab&ac\\ba&-b^2&bc\\ca&cb&-c^2\end{vmatrix}\)=4a²b²c²

Answer 1

△=\(\begin{vmatrix}-a^2&ab&ac\\ba&-b^2&bc\\ca&cb&-c^2\end{vmatrix}\)

=\(abc\begin{vmatrix}-a&b&c\\a&-b&c\\a&b&-c\end{vmatrix}\)       [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