Using the property of determinants and without expanding, prove that:`|[-a^2,a b, a c],[ b a, -b^2,b c],[c a, c b,-c^2]|=4a^2b^2c^2`

Explanation

`L.H.S. = |[-a^2,ab,ac],[ba,-b^2,bc],[ca,cb,-c^2]|`

`=abc|[-a,b,c],[a,-b,c],[a,b,-c]|`(Taking `a,b,c` common along rows)

`=a^2b^2c^2|[-1,1,1],[1,-1,1],[1,1,-1]|`(Taking `a,b,c` common along columns)

Applying `R_2->R_2+R_1` and `R_3->R_3+R_1`

`=a^2b^2c^2|[-1,1,1],[0,0,2],[0,2,0]|`

`=a^2b^2c^2[-1(0-4)-1(0)+1(0)]`

`=4a^2b^2c^2 = R.H.S.`