If the equation x pmatrix 1 2 3 pmatrix + y pmatrix 2 3 4 pmatrix + z pmatrix 1 1 a pmatrix = pmatrix 4 7 b pmatrix has infinitely many solutions for x, y, z R , then the value of a+b is equal to

Step-by-step solution

The given matrix equation can be written as a system of linear equations: x + 2y + z = 4 2x + 3y + z = 7 3x + 4y + az = b For the system to have infinitely many solutions, the determinant of the coefficient matrix must be zero: = vmatrix 1 & 2 & 1 2 & 3 & 1 3 & 4 & a vmatrix = 0 Expanding along the first row: 1(3a - 4) - 2(2a - 3) + 1(8 - 9) = 0 3a - 4 - 4a + 6 - 1 = 0 -a + 1 = 0 a = 1 Substituting a = 1 into the third equation, the system becomes: x + 2y + z = 4 (i) 2x + 3y + z = 7 (ii) 3x + 4y + z = b (iii) Obser