The value of $\begin{bmatrix}a^2 & -ab & -ac\\-ab & b^2 & -bc\\ca & bc & -c^2\end{bmatrix},$ is

$4a^2b^2c^2$

The correct answer is option (4) : $4a^2b^2c^2$

We have,

$Δ=\begin{bmatrix}a^2 & -ab & -ac\\-ab & b^2 & -bc\\ca & bc & -c^2\end{bmatrix}$

$⇒Δ= abc\begin{bmatrix}a & -b & -c\\-a & b & -c\\a & b & -c\end{bmatrix}$                  $\begin{matrix} Taking \,\, a, b, c \, common\, from\\ R_1,\,  R_2 \, and \, R_3 \, respectively \end{matrix}$

$⇒Δ=a^2b^2c^2\begin{bmatrix}1& -1 & -1\\-1 & 1 & -1\\1 & 1 & -1\end{bmatrix}$           $\begin{matrix} Taking \,\, a, b, c \, common\, from\\ C_1,\,  C_2 \, and \, C_3 \, respectively \end{matrix}$

$⇒Δ=a^2b^2c^2\begin{bmatrix}1& 0 & 0\\-1 & 0 & -2\\1 &2 & 0\end{bmatrix}$            $\begin{matrix}  Applying \,\, C_2→C_2+C_1,\\C_3→C_3+C_1\end{matrix}$

$⇒Δ=4a^2b^2c^2$