Consider the augmented matrix

\[ \left[\begin{array}{ccc|c} 1 & -1 & -1 & -6 \\ 0 & 1 & -3 & 4 \\ 0 & 0 & 0 & 1 \\ 0 & 2 & -6 & 7 \end{array}\right] \]

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

Answer:

  1. \[\begin{matrix} x_{1} & - & x_{2} & - & x_{3} & = & -6 \\ & & x_{2} & - & 3 \, x_{3} & = & 4 \\ & & & & 0 & = & 1 \\ & & 2 \, x_{2} & - & 6 \, x_{3} & = & 7 \\ \end{matrix}\]

  2. \[ x_{1} \left[\begin{array}{c} 1 \\ 0 \\ 0 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} -1 \\ 1 \\ 0 \\ 2 \end{array}\right] + x_{3} \left[\begin{array}{c} -1 \\ -3 \\ 0 \\ -6 \end{array}\right] = \left[\begin{array}{c} -6 \\ 4 \\ 1 \\ 7 \end{array}\right] \]