Question

Prove that `|(-a^2,ab,ac),(ab,-b^2,bc),(ac,bc,-c^2)| = 4a^2b^2c^2`

Answer 1

`|(-a^2,ab,ac),(ab,-b^2,bc),(ac,bc,-c^2)|`

= `abc|(-a,b,c),(a,-b,c),(a,b,-c)|`   ....(Take out a,b,c respectively from R₁, R₂ and R₃)

= `a^2b^2c^2 |(-1,1,1),(1,-1,1),(1,1,-1)|`   ...(Take out a,b,c respectively from C₁, C₂ and C₃)

= `a^2b^2c^2 |(-1,1,1),(0,0,2),(0,2,0)|`  ...`[("R"_2 -> "R"_2 + "R"_1),("R"_3 -> "R"_3 + "R"_1)]`

= a²b²c² [-(0 – 4) + 0 + 0]

= 4a²b²c²