The value of \(\left|\begin{array}{ccc} -a^{2} & a b & a c \\ a b & -b^{2} & b c \\ a c & b c & -c^{2} \end{array}\right| \) is : 

Concept Used:

Properties of determinants.

Factor out common terms from rows or columns.

Calculation:

Given:

Δ = \(\begin{vmatrix} -a^2 & ab & ac \\ ab & -b^2 & bc \\ ac & bc & -c^2 \end{vmatrix}\)

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

Δ = abc \(\begin{vmatrix} -a & b & c \\ a & -b & c \\ a & b & -c \end{vmatrix}\) = a²b²c² \(\begin{vmatrix} -1 & 1 & 1 \\ 1 & -1 & 1 \\ 1 & 1 & -1 \end{vmatrix}\)

[taking a, b, c common from C₁, C₂, C₃ respectively]

Δ = a²b²c² \(\begin{vmatrix} -1 & 0 & 0 \\ 1 & 0 & 2 \\ 1 & 2 & 0 \end{vmatrix}\) (applying C₂ → C₂ + C₁, C₃ → C₃ + C₁)

Δ = a²b²c² (-1) \(\begin{vmatrix} 0 & 2 \\ 2 & 0 \end{vmatrix}\) = a²b²c² (-1) (0 - 4)

Δ = 4a²b²c²

Hence option 3 is correct