Prove that |(-a^2,ab,ac),(ab,-b^2,bc),(ac,bc,-c^2)| = 4a^2b^2c^2.

Question

Prove that \(\begin{vmatrix}-a^2 &ab&ac \\[0.3em]ab&-b^2&bc\\ac&bc&-c^2\end{vmatrix}\) = 4a²b²c².

Answer 1

LHS = \(\begin{vmatrix}-a^2 &ab&ac \\[0.3em]ab&-b^2&bc\\ac&bc&-c^2\end{vmatrix}\)

Taking a from R₁, b from R₂ and c from R₃ as common factors we get,