Solve the Problem $A = \left[ \begin{array} { r r r } 1 & - 3 & 2 \\ - 2 & 5 & - 1 \\ 3 & - 4 & 5 \end{array} \right]$

Multiple Choice

Solve the problem.
-Let $A = \left[ \begin{array} { r r r } 1 & - 3 & 2 \\ - 2 & 5 & - 1 \\ 3 & - 4 & 5 \end{array} \right]$ and $\mathbf { b } = \left[ \begin{array} { l } b _ { 1 } \\ b _ { 2 } \\ b _ { 3 } \end{array} \right]$ .
Determine if the equation $\mathrm { Ax } = \mathrm { b }$ is consistent for all possible $\mathrm { b } _ { 1 } , \mathrm {~b} _ { 2 } , \mathrm {~b} _ { 3 }$ . If the equation is not consistent for all possible $b _ { 1 } , b _ { 2 } , b _ { 3 }$ , give a description of the set of all $\mathbf { b }$ for which the equation is consistent (i.e., a condition which must be satisfied by $b _ { 1 } , b _ { 2 } , b _ { 3 }$ ) .

A) Equation is consistent for all $b _ { 1 } , b _ { 2 } , b _ { 3 }$ satisfying $- 3 b _ { 1 } + b _ { 3 } = 0$ .
B) Equation is consistent for all $b _ { 1 } , b _ { 2 } , b _ { 3 }$ satisfying $2 b _ { 1 } + b _ { 2 } = 0$ .
C) Equation is consistent for all possible $b _ { 1 } , b _ { 2 } , b _ { 3 }$ .
D) Equation is consistent for all $b _ { 1 } , b _ { 2 }$ , $b _ { 3 }$ satisfying $7 b _ { 1 } + 5 b _ { 2 } + b _ { 3 } = 0$ .