Consider the augmented matrix

\[ \left[\begin{array}{ccc|c} 4 & 2 & 0 & -6 \\ 1 & 2 & -5 & 0 \\ 5 & 4 & -6 & -6 \\ -3 & -1 & -2 & 5 \end{array}\right] \]

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

Answer:

  1. \[\begin{matrix} 4 \, x & + & 2 \, y & & & = & -6 \\ x & + & 2 \, y & - & 5 \, z & = & 0 \\ 5 \, x & + & 4 \, y & - & 6 \, z & = & -6 \\ -3 \, x & - & y & - & 2 \, z & = & 5 \\ \end{matrix}\]

  2. \[ x_{1} \left[\begin{array}{c} 4 \\ 1 \\ 5 \\ -3 \end{array}\right] + x_{2} \left[\begin{array}{c} 2 \\ 2 \\ 4 \\ -1 \end{array}\right] + x_{3} \left[\begin{array}{c} 0 \\ -5 \\ -6 \\ -2 \end{array}\right] = \left[\begin{array}{c} -6 \\ 0 \\ -6 \\ 5 \end{array}\right] \]