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Q.
The value of $\begin{vmatrix}- a ^{2} & ab & ac \\ ab & - b ^{2} & bc \\ ac & bc & - c ^{2}\end{vmatrix}$ is
Determinants
Solution:
Let $\Delta=\begin{vmatrix}-a^{2} & a b & a c \\ a b & -b^{2} & b c \\ a c & b c & -c^{2}\end{vmatrix}$
Taking $a , b , c$ common from $R _{1}, R _{2}$ and $R _{3}$ respectively, we get. $\Delta=a b c\begin{vmatrix}-a & b & c \\ a & -b & c \\ a & b & -c\end{vmatrix}]=a^{2} b^{2} c^{2}[\begin{vmatrix}-1 & 1 & 1 \\ 1 & -1 & 1 \\ 1 & 1 & -1\end{vmatrix}$
taking a, b, c common from $C _{1}, C _{2}, C _{3}$ respectively]
$=a^{2} b^{2} c^{2} \begin{vmatrix}-1 & 0 & 0 \\ 1 & 0 & 2 \\ 1 & 2 & 0\end{vmatrix}$
( applying $C _{2} \rightarrow C _{2}+ C _{1}, C _{3} \rightarrow C _{3}+ C _{1})$
$=a^{2} b^{2} c^{2} \cdot(-1)\begin{vmatrix}0 & 2 \\ 2 & 0\end{vmatrix}|=a^{2} b^{2} c^{2}(-1)(0-4)$
$\Rightarrow \Delta=4 a^{2} b^{2} c^{2}$