Solve the Problem , And $\mathbf { b } = \left[ \begin{array} { r } - 1 \\ 1 \\ 2 \end{array} \right]$

Multiple Choice

Solve the problem.
-Let $\mathbf { a } _ { \mathbf { 1 } } = \left[ \begin{array} { r } 1 \\ 2 \\ - 3 \end{array} \right] , \mathbf { a } _ { \mathbf { 2 } } = \left[ \begin{array} { r } - 3 \\ - 4 \\ 1 \end{array} \right] , \mathbf { a } _ { 3 } = \left[ \begin{array} { l } 2 \\ 1 \\ 6 \end{array} \right]$ , and $\mathbf { b } = \left[ \begin{array} { r } - 1 \\ 1 \\ 2 \end{array} \right]$ .
Determine whether $\mathbf { b }$ can be written as a linear combination of $\mathbf { a } _ { \mathbf { 1 } } , \mathbf { a } _ { \mathbf { 2 } }$ , and $\mathbf { a } _ { 3 }$ . In other words, determine whether weights $x _ { 1 } , x _ { 2 }$ , and $x _ { 3 }$ exist, such that $x _ { 1 } a _ { 1 } + x _ { 2 } a _ { 2 } + x _ { 3 } a _ { 3 } = b$ . Determine the weights $x _ { 1 } , x _ { 2 }$ , and $x _ { 3 }$ if possible.

A) $x _ { 1 } = 2 , x _ { 2 } = 1 , x _ { 3 } = 0$
B) $x _ { 1 } = - 3 , x _ { 2 } = 0 , x _ { 3 } = 1$
C) No solution
D) $x _ { 1 } = - 2 , x _ { 2 } = - 1 , x _ { 3 } = 1$