Consider a 3 3 non-singular matrix A= [a_ i j ] . A matrix B= [b_ i j ]_ 3 3 is formed such that b_ i j is the sum of all the elements of i^ t h row in A except a_ i j . If there exists a matrix C such that A C=B , then:

Step-by-step solution

Let A= [ array lll a₁₁ & a₁₂ & a₁₃ a₂₁ & a₂₂ & a₂₃ a₃₁ & a₃₂ & a₃₃ array ] and B= [ array lll a₁₂+a₁₃ & a₁₁+a₁₃ & a₁₁+a₁₂ a₂₂+a₂₃ & a₂₁+a₂₃ & a₂₁+a₂₂ a₃₂+a₃₃ & a₃₁+a₃₃ & a₃₁+a₃₂ array ] array ll & C= [ array lll 0 & 1 & 1 1 & 0 & 1 1 & 1 & 0 array ] & C is symmetric matrix. & |C|=0-1(0-1)+1(1-0)=2 & |B|=|C||A|=2|A| array