Let A = pmatrix 1 & 2 & 1 2 & & 3 1 & 2 & pmatrix and B = pmatrix 3 9 pmatrix . If the system of equations AX = B has infinitely many solutions, then the value of + + trace ( adj (A)) is equal to

Step-by-step solution

For the system AX = B to have infinitely many solutions, we must have (A) = 0 . (A) = vmatrix 1 & 2 & 1 2 & & 3 1 & 2 & vmatrix = 0 Expanding along the first row: 1( ^2 - 6) - 2(2 - 3) + 1(4 - ) = 0 ^2 - 6 - 4 + 6 + 4 - = 0 ^2 - 5 + 4 = 0 ( - 1)( - 4) = 0 = 1 or = 4 If = 1 , the first and third rows of the augmented matrix correspond to the equations x + 2y + z = 3 and x + 2y + z = 9 , which are inconsistent. Thus, = 4 . For = 4 , the system is: x + 2y + z = 3 (i) 2x + 4y + 3z = (ii) x + 2y + 4z = 9 (iii) Subtracti