Suppose p, q, r 0 and system of equation gathered (p+a) x+b y+c z=0 a x+(q+b) y+c z=0 gathered a x+b y+(r+c) z=0 , has a non-trivial solution, then the value of a p + b q + c r is

Step-by-step solution

Let |A|= | array ccc p+a & b & c a & q+b & c a & b & r+c array | Also, given that equations has non trivial solution |A|=0 aligned & | array ccc p+a & b & c a & q+b & c a & b & r+c array |=0 & (p+a)[(q+b)(r+c)-b c]-b[a(r+c)-c a] & +c[a b-a(q+b)]=0 & (p+a)[q r+q c+b r]-b[a r]+c[-a q]=0 & p q r+p q c+p b r+a q r=0 aligned Dividing whole equation by p q r aligned & p q r p q r + p q c p q r + p r b p q r + q r a p q r & =0 & 1+ c r + b q + a p & =0 & a p + b q + c r & =-1 aligned