Let M be a 3 3 matrix such that M pmatrix 1 0 0 pmatrix = pmatrix 1 2 3 pmatrix , M pmatrix 0 1 0 pmatrix = pmatrix 0 1 2 pmatrix and M pmatrix 0 0 1 pmatrix = pmatrix -1 1 1 pmatrix . If M pmatrix x y z pmatrix = pmatrix 1 7 11 pmatrix , then x + y + z equals :

Step-by-step solution

Let M be the matrix whose columns are the given vectors. We have: M = pmatrix 1 & 0 & -1 2 & 1 & 1 3 & 2 & 1 pmatrix Given M pmatrix x y z pmatrix = pmatrix 1 7 11 pmatrix , we can write this as a system of linear equations: x - z = 1 2x + y + z = 7 3x + 2y + z = 11 Subtracting the second equation from the third equation gives: x + y = 4 Adding the first equation and the second equation gives: 3x + y = 8 Subtracting the equation x + y = 4 from 3x + y = 8 gives: 2x = 4 x = 2 Substituting x = 2 into x + y = 4 gives: