Consider the vector equation.

\[ x_{1} \left[\begin{array}{c} 1 \\ 1 \\ -1 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} -3 \\ -3 \\ 3 \\ 0 \end{array}\right] + x_{3} \left[\begin{array}{c} -4 \\ -3 \\ 3 \\ -2 \end{array}\right] = \left[\begin{array}{c} 6 \\ 4 \\ -4 \\ 4 \end{array}\right] \]

  1. Write a system of scalar equations corresponding to this vector equation.
  2. Write an augmented matrix corresponding to this vector equation.

Answer:

  1. \[\begin{matrix} x_{1} & - & 3 \, x_{2} & - & 4 \, x_{3} & = & 6 \\ x_{1} & - & 3 \, x_{2} & - & 3 \, x_{3} & = & 4 \\ -x_{1} & + & 3 \, x_{2} & + & 3 \, x_{3} & = & -4 \\ & & & & 2 \, x_{3} & = & 4 \\ \end{matrix}\]

  2. \[ \left[\begin{array}{ccc|c} 1 & -3 & -4 & 6 \\ 1 & -3 & -3 & 4 \\ -1 & 3 & 3 & -4 \\ 0 & 0 & -2 & 4 \end{array}\right] \]