Let M be a $3 \times 3$ matrix such that
$M \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}, M \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix}$ and $M \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} -1 \\ 1 \\ 1 \end{pmatrix}$. If $M \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 1 \\ 7 \\ 11 \end{pmatrix}$, then $x+y+z$ equals :
- 4
- 5
- 7
- 11
The Correct Option is C
Solution and Explanation
Step 1: Understanding the Concept:
We can express the unknown vector \([x, y, z]^T\) as a linear combination of vectors for which the outputs of \(M\) are known. By linearity: \[ M(a\vec{u} + b\vec{v} + c\vec{w}) = aM\vec{u} + bM\vec{v} + cM\vec{w} \]
Step 2: Key Formula or Approach:
Let \[ \vec{v_1} = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}, \quad \vec{v_2} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \quad \vec{v_3} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \] Assume: \[ \begin{pmatrix} x \\ y \\ z \end{pmatrix} = a\vec{v_1} + b\vec{v_2} + c\vec{v_3} = \begin{pmatrix} a \\ a+b \\ c \end{pmatrix} \] Thus: \[ x = a,\quad y = a+b,\quad z = c \] \[ x + y + z = 2a + b + c \]
Step 3: Detailed Explanation:
Applying matrix \(M\): \[ M \begin{pmatrix} x \\ y \\ z \end{pmatrix} = aM\vec{v_1} + bM\vec{v_2} + cM\vec{v_3} = \begin{pmatrix} 1 \\ 7 \\ 11 \end{pmatrix} \] \[ a \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + b \begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix} + c \begin{pmatrix} -1 \\ 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \\ 7 \\ 11 \end{pmatrix} \] This gives equations: \[ a - c = 1 \] \[ 2a + b + c = 7 \] \[ 3a + 2b + c = 11 \] From first: \[ c = a - 1 \] Substitute: \[ 2a + b + a - 1 = 7 \Rightarrow 3a + b = 8 \quad (iv) \] \[ 3a + 2b + a - 1 = 11 \Rightarrow 4a + 2b = 12 \Rightarrow 2a + b = 6 \quad (v) \] Subtract: \[ (3a + b) - (2a + b) = 8 - 6 \Rightarrow a = 2 \] Then: \[ b = 8 - 3a = 2,\quad c = a - 1 = 1 \] Thus: \[ x = 2,\quad y = 4,\quad z = 1 \] \[ x + y + z = 2 + 4 + 1 = 7 \]
Step 4: Final Answer:
The sum \(x + y + z = 7\).
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