Let M be a 3 3 matrix satisfying M [ array l 0 1 0 array ]= [ array c -1 2 3 array ], M [ array c 1 -1 0 array ]= [ array c 1 1 -1 array ] and M [ array l 1 1 1 array ]= [ array c 0 0 12 array ] Then, the sum of the diagonal entries of M is

Step-by-step solution

Let M= [ array lll a₁ & a₂ & a₃ b₁ & b₂ & b₃ c₁ & c₂ & c₃ array ] M [ array l 0 1 0 array ]= [ array r -1 2 3 array ], M [ array r 1 -1 0 array ]= [ array r 1 1 -1 array ] , M [ array l 1 1 1 array ]= [ array r 0 0 12 array ] , aligned & [ array l a₂ b₂ c₂ array ]= [ array r -1 2 3 array ], [ array l a₁-a₂ b₁-b₂ c₁-c₂ array ]= [ array r 1 1 -1 array ], & [ array l a₁+a₂+a₃ b₁+b₂+b₃ c₁+c₂+c₃ array ]= [ array r 0 0 12 array ] & a₂=-1, b₂=2, c₂=3, a₁-a₂=1, & b₁-b₂=1, c₁-c₂=-1 & a₁+a₂+a₃=0, b₁+b₂+b₃=0, & c₁+c₂+c₃=12 &