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 :
B. 5
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: 2 + y = 4 y = 2 Substituting x = 2 into x - z = 1 gives: 2 - z = 1 z = 1 Therefore, x + y + z = 2 + 2 + 1 = 5. Answer: 5
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