Question

Let the given determinant be ( Delta ). Then, ( Delta=left|begin{array}{ccc}-a^{2} & a b & a c a b & -b^{2} & b c a c & b c & -c^{2}end{array}right| )
( =(a b c) cdotleft|begin{array}{ccc}-a & a & a b & -b & b c & c & -cend{array}right|left[begin{array}{c}text { taking out } a, b, c text { common from } C_{1}, C_{2}, C_{3} text { respectively }end{array}right] )
( =left|begin{array}{ccc}-a & 0 & 0 b & 0 & 2 b c & 2 c & 0end{array}right| quadleft[C_{2} rightarrowleft(C_{2}+C_{1}right) text { and } C_{3} rightarrowleft(C_{3}+C_{1}right)right] )
( =(a b c) cdot(-a)left|begin{array}{cc}0 & 2 b 2 c & 0end{array}right|=(a b c)(-a)(-4 b c) )
( =4 a^{2} b^{2} c^{2} )

# L-a? ab ac EXAMPLE 4 Prove that ab-b2 bc = 4a2b2c2. ac bc -

Solution