નિશ્ચાયકના ગુણધર્મનો ઉપયોગ કરીને સાબિત કરો : $\left|\begin{array}{ccc}-a^{2} & a b & a c \\ b a & -b^{2} & b c \\ c a & c b & -c^{2}\end{array}\right|=4 a^{2} b^{2} c^{2}$

$\Delta=\left|\begin{array}{ccc}-a^{2} & a b & a c \\ b a & -b^{2} & b c \\ c a & c b & -c^{2}\end{array}\right|$

$=a b c\left|\begin{array}{ccc}-a & b & c \\ a & -b & c \\ a & b & -c\end{array}\right|$ 

Taking out factors $ a,\  b,\  c $ from $ R_{1}, R_{2}$  and $R_{3}$

$=a^{2} b^{2} c^{2}\left|\begin{array}{ccc}-1 & 1 & 1 \\ 1 & -1 & 1 \\ 1 & 1 & -1\end{array}\right| $

[ Taking out factors $ a,\  b,\  c$ from $ C_{1}, C_{2} $ and $C_{3}]$

Applying $R_{2} \rightarrow R_{2}+R_{1}$ and $R_{3} \rightarrow R_{3}+R_{1},$ we have:

$\Delta=a^{2} b^{2} c^{2}\left|\begin{array}{ccc}-1 & 1 & 1 \\ 0 & 0 & 2 \\ 0 & 2 & 0\end{array}\right|$

$ = {a^2}{b^2}{c^2}( - 1)\left| {\begin{array}{*{20}{c}}
  0&2 \\ 
  2&0 
\end{array}} \right|$

$ =  - {a^2}{b^2}{c^2}(0 - 4) = 4{a^2}{b^2}{c^2}$