If p+q+r=a+b+c=0 , then the determinant | array lll p a & q b & r c q c & ra & p b r b & p c & q a array | equals

Step-by-step solution

| array ccc pa & qb & rc qc & ra & pb rb & pc & qa array |= pa ( rqa ^2- p ^2 bc )- qb ( q ^2 ac - prb ^2 )+ aligned & rc (q qcc ^2- r ^2 a b ) & = pqra ^3- p ^3 a b c- q ^3 a b c+ pqrb ^3+ pqrc ^3- r ^3 a b c & = pqr ( a ^3+ b ^3+ c ^3 )- abc ( p ^3+ q ^3+ r ^3 ) & Given, p + q + r = a + b + c =0 aligned aligned & p ^3+ q ^3+ r ^3=3 pqr and a ^3+ b ^3+ c ^3=3 abc . & pqr ( a ^3+ b ^3+ c ^3 )- abc ( p ^3+ q ^3+ r ^3 ) & = pqr (3 abc )- abc (3 pqr ) & =0 aligned