Consider the augmented matrix

\[ \left[\begin{array}{ccc|c} -1 & -3 & 5 & 3 \\ 0 & 1 & -1 & -2 \\ 2 & 3 & -7 & 0 \\ 1 & 1 & -3 & 1 \end{array}\right] \]

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

Answer:

  1. \[\begin{matrix} -x_{1} & - & 3 \, x_{2} & + & 5 \, x_{3} & = & 3 \\ & & x_{2} & - & x_{3} & = & -2 \\ 2 \, x_{1} & + & 3 \, x_{2} & - & 7 \, x_{3} & = & 0 \\ x_{1} & + & x_{2} & - & 3 \, x_{3} & = & 1 \\ \end{matrix}\]

  2. \[ x_{1} \left[\begin{array}{c} -1 \\ 0 \\ 2 \\ 1 \end{array}\right] + x_{2} \left[\begin{array}{c} -3 \\ 1 \\ 3 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} 5 \\ -1 \\ -7 \\ -3 \end{array}\right] = \left[\begin{array}{c} 3 \\ -2 \\ 0 \\ 1 \end{array}\right] \]