E1 - Linear systems, vector equations, and augmented matrices


Example 1 πŸ”—

Consider the vector equation.

\[ x_{1} \left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} 0 \\ 1 \\ 2 \end{array}\right] + x_{3} \left[\begin{array}{c} 2 \\ 0 \\ 1 \end{array}\right] + x_{4} \left[\begin{array}{c} 6 \\ -2 \\ -2 \end{array}\right] = \left[\begin{array}{c} -6 \\ -3 \\ -8 \end{array}\right] \]

  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{alignat*}{5} x & & &+& 2 \, z &+& 6 \, {w} &=& -6 \\ & & y & & &-& 2 \, {w} &=& -3 \\ & & 2 \, y &+& z &-& 2 \, {w} &=& -8 \\ \end{alignat*}
  2. \[ \left[\begin{array}{cccc|c} 1 & 0 & 2 & 6 & -6 \\ 0 & 1 & 0 & -2 & -3 \\ 0 & 2 & 1 & -2 & -8 \end{array}\right] \]


Example 2 πŸ”—

Consider the system of equations \begin{alignat*}{4} x & & &-& 3 \, z &=& 4 \\ & & y &-& 3 \, z &=& -2 \\2 \, x &+& y &-& 8 \, z &=& 6 \\x &-& y &-& 3 \, z &=& 6 \\ \end{alignat*}

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

Answer:

  1. \[ \left[\begin{array}{ccc|c} 1 & 0 & -3 & 4 \\ 0 & 1 & -3 & -2 \\ 2 & 1 & -8 & 6 \\ 1 & -1 & -3 & 6 \end{array}\right] \]

  2. \[ x_{1} \left[\begin{array}{c} 1 \\ 0 \\ 2 \\ 1 \end{array}\right] + x_{2} \left[\begin{array}{c} 0 \\ 1 \\ 1 \\ -1 \end{array}\right] + x_{3} \left[\begin{array}{c} -3 \\ -3 \\ -8 \\ -3 \end{array}\right] = \left[\begin{array}{c} 4 \\ -2 \\ 6 \\ 6 \end{array}\right] \]


Example 3 πŸ”—

Consider the vector equation.

\[ x_{1} \left[\begin{array}{c} 1 \\ 0 \\ -2 \end{array}\right] + x_{2} \left[\begin{array}{c} 0 \\ 1 \\ 0 \end{array}\right] + x_{3} \left[\begin{array}{c} 2 \\ 4 \\ -4 \end{array}\right] + x_{4} \left[\begin{array}{c} -3 \\ 1 \\ 6 \end{array}\right] = \left[\begin{array}{c} 4 \\ -3 \\ -8 \end{array}\right] \]

  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{alignat*}{5} x_{1} & & &+& 2 \, x_{3} &-& 3 \, x_{4} &=& 4 \\ & & x_{2} &+& 4 \, x_{3} &+& x_{4} &=& -3 \\-2 \, x_{1} & & &-& 4 \, x_{3} &+& 6 \, x_{4} &=& -8 \\ \end{alignat*}
  2. \[ \left[\begin{array}{cccc|c} 1 & 0 & 2 & -3 & 4 \\ 0 & 1 & 4 & 1 & -3 \\ -2 & 0 & -4 & 6 & -8 \end{array}\right] \]


Example 4 πŸ”—

Consider the vector equation.

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

  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{alignat*}{4} x &-& y &-& 5 \, z &=& 3 \\-x &-& 4 \, y &-& 5 \, z &=& 2 \\-x &-& y &+& z &=& -1 \\ &-& 2 \, y &-& 4 \, z &=& 2 \\ \end{alignat*}
  2. \[ \left[\begin{array}{ccc|c} 1 & -1 & -5 & 3 \\ -1 & -4 & -5 & 2 \\ -1 & -1 & 1 & -1 \\ 0 & -2 & -4 & 2 \end{array}\right] \]


Example 5 πŸ”—

Consider the augmented matrix

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

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


Example 6 πŸ”—

Consider the augmented matrix

\[ \left[\begin{array}{ccc|c} -2 & 3 & -1 & -7 \\ 1 & -2 & 0 & 4 \\ -2 & 1 & -2 & -4 \\ 2 & -6 & -7 & 5 \end{array}\right] \]

  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{alignat*}{4} -2 \, x_{1} &+& 3 \, x_{2} &-& x_{3} &=& -7 \\x_{1} &-& 2 \, x_{2} & & &=& 4 \\-2 \, x_{1} &+& x_{2} &-& 2 \, x_{3} &=& -4 \\2 \, x_{1} &-& 6 \, x_{2} &-& 7 \, x_{3} &=& 5 \\ \end{alignat*}
  2. \[ x_{1} \left[\begin{array}{c} -2 \\ 1 \\ -2 \\ 2 \end{array}\right] + x_{2} \left[\begin{array}{c} 3 \\ -2 \\ 1 \\ -6 \end{array}\right] + x_{3} \left[\begin{array}{c} -1 \\ 0 \\ -2 \\ -7 \end{array}\right] = \left[\begin{array}{c} -7 \\ 4 \\ -4 \\ 5 \end{array}\right] \]


Example 7 πŸ”—

Consider the vector equation.

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

  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{alignat*}{4} x_{1} &+& 3 \, x_{2} &-& 5 \, x_{3} &=& 3 \\3 \, x_{1} &+& x_{2} &-& 6 \, x_{3} &=& -6 \\ &-& x_{2} &+& x_{3} &=& -2 \\ &-& 2 \, x_{2} & & &=& -6 \\ \end{alignat*}
  2. \[ \left[\begin{array}{ccc|c} 1 & 3 & -5 & 3 \\ 3 & 1 & -6 & -6 \\ 0 & -1 & 1 & -2 \\ 0 & -2 & 0 & -6 \end{array}\right] \]


Example 8 πŸ”—

Consider the vector equation.

\[ x_{1} \left[\begin{array}{c} 4 \\ 3 \\ 2 \end{array}\right] + x_{2} \left[\begin{array}{c} -3 \\ -2 \\ -3 \end{array}\right] + x_{3} \left[\begin{array}{c} -2 \\ -1 \\ -3 \end{array}\right] + x_{4} \left[\begin{array}{c} -8 \\ -7 \\ 0 \end{array}\right] = \left[\begin{array}{c} 1 \\ -1 \\ 8 \end{array}\right] \]

  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{alignat*}{5} 4 \, x &-& 3 \, y &-& 2 \, z &-& 8 \, {w} &=& 1 \\3 \, x &-& 2 \, y &-& z &-& 7 \, {w} &=& -1 \\2 \, x &-& 3 \, y &-& 3 \, z & & &=& 8 \\ \end{alignat*}
  2. \[ \left[\begin{array}{cccc|c} 4 & -3 & -2 & -8 & 1 \\ 3 & -2 & -1 & -7 & -1 \\ 2 & -3 & -3 & 0 & 8 \end{array}\right] \]


Example 9 πŸ”—

Consider the augmented matrix

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

  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{alignat*}{5} x &-& 2 \, y & & &+& 3 \, {w} &=& 3 \\ & & y &+& 2 \, z &+& 2 \, {w} &=& -5 \\x &-& 2 \, y &+& z &+& 5 \, {w} &=& 1 \\ \end{alignat*}
  2. \[ x_{1} \left[\begin{array}{c} 1 \\ 0 \\ 1 \end{array}\right] + x_{2} \left[\begin{array}{c} -2 \\ 1 \\ -2 \end{array}\right] + x_{3} \left[\begin{array}{c} 0 \\ 2 \\ 1 \end{array}\right] + x_{4} \left[\begin{array}{c} 3 \\ 2 \\ 5 \end{array}\right] = \left[\begin{array}{c} 3 \\ -5 \\ 1 \end{array}\right] \]


Example 10 πŸ”—

Consider the system of equations \begin{alignat*}{5} x &-& y &+& 3 \, z &-& 4 \, {w} &=& 0 \\ & & y &-& 4 \, z &+& 2 \, {w} &=& -1 \\ &-& y &+& 4 \, z &-& 2 \, {w} &=& 1 \\ \end{alignat*}

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

Answer:

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

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


Example 11 πŸ”—

Consider the vector equation.

\[ x_{1} \left[\begin{array}{c} 2 \\ 1 \\ 4 \end{array}\right] + x_{2} \left[\begin{array}{c} 3 \\ 2 \\ 4 \end{array}\right] + x_{3} \left[\begin{array}{c} -3 \\ -1 \\ -8 \end{array}\right] + x_{4} \left[\begin{array}{c} 7 \\ 5 \\ 8 \end{array}\right] = \left[\begin{array}{c} 4 \\ 3 \\ 4 \end{array}\right] \]

  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{alignat*}{5} 2 \, x &+& 3 \, y &-& 3 \, z &+& 7 \, {w} &=& 4 \\x &+& 2 \, y &-& z &+& 5 \, {w} &=& 3 \\4 \, x &+& 4 \, y &-& 8 \, z &+& 8 \, {w} &=& 4 \\ \end{alignat*}
  2. \[ \left[\begin{array}{cccc|c} 2 & 3 & -3 & 7 & 4 \\ 1 & 2 & -1 & 5 & 3 \\ 4 & 4 & -8 & 8 & 4 \end{array}\right] \]


Example 12 πŸ”—

Consider the augmented matrix

\[ \left[\begin{array}{cccc|c} 1 & 0 & -2 & 1 & 1 \\ 3 & 1 & -8 & 0 & 2 \\ 2 & 1 & -6 & -1 & 1 \end{array}\right] \]

  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{alignat*}{5} x_{1} & & &-& 2 \, x_{3} &+& x_{4} &=& 1 \\3 \, x_{1} &+& x_{2} &-& 8 \, x_{3} & & &=& 2 \\2 \, x_{1} &+& x_{2} &-& 6 \, x_{3} &-& x_{4} &=& 1 \\ \end{alignat*}
  2. \[ x_{1} \left[\begin{array}{c} 1 \\ 3 \\ 2 \end{array}\right] + x_{2} \left[\begin{array}{c} 0 \\ 1 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} -2 \\ -8 \\ -6 \end{array}\right] + x_{4} \left[\begin{array}{c} 1 \\ 0 \\ -1 \end{array}\right] = \left[\begin{array}{c} 1 \\ 2 \\ 1 \end{array}\right] \]


Example 13 πŸ”—

Consider the augmented matrix

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

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


Example 14 πŸ”—

Consider the vector equation.

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

  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{alignat*}{4} x_{1} &+& x_{2} &+& 4 \, x_{3} &=& 1 \\ & & x_{2} &-& x_{3} &=& 2 \\ &-& 4 \, x_{2} &+& 4 \, x_{3} &=& -7 \\ & & 4 \, x_{2} &-& 4 \, x_{3} &=& 7 \\ \end{alignat*}
  2. \[ \left[\begin{array}{ccc|c} 1 & 1 & 4 & 1 \\ 0 & 1 & -1 & 2 \\ 0 & -4 & 4 & -7 \\ 0 & 4 & -4 & 7 \end{array}\right] \]


Example 15 πŸ”—

Consider the augmented matrix

\[ \left[\begin{array}{ccc|c} -1 & -3 & 2 & 1 \\ 1 & 3 & -4 & -3 \\ 0 & 0 & 0 & 0 \\ 2 & 6 & -7 & -5 \end{array}\right] \]

  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{alignat*}{4} -x_{1} &-& 3 \, x_{2} &+& 2 \, x_{3} &=& 1 \\x_{1} &+& 3 \, x_{2} &-& 4 \, x_{3} &=& -3 \\ & & & & 0 &=& 0 \\2 \, x_{1} &+& 6 \, x_{2} &-& 7 \, x_{3} &=& -5 \\ \end{alignat*}
  2. \[ x_{1} \left[\begin{array}{c} -1 \\ 1 \\ 0 \\ 2 \end{array}\right] + x_{2} \left[\begin{array}{c} -3 \\ 3 \\ 0 \\ 6 \end{array}\right] + x_{3} \left[\begin{array}{c} 2 \\ -4 \\ 0 \\ -7 \end{array}\right] = \left[\begin{array}{c} 1 \\ -3 \\ 0 \\ -5 \end{array}\right] \]


Example 16 πŸ”—

Consider the system of equations \begin{alignat*}{4} 6 \, x &+& 5 \, y &-& 7 \, z &=& -4 \\-5 \, x &-& 4 \, y &+& 6 \, z &=& 4 \\5 \, x &+& 3 \, y &-& 6 \, z &=& -7 \\-5 \, x &-& 3 \, y &+& 7 \, z &=& 8 \\ \end{alignat*}

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

Answer:

  1. \[ \left[\begin{array}{ccc|c} 6 & 5 & -7 & -4 \\ -5 & -4 & 6 & 4 \\ 5 & 3 & -6 & -7 \\ -5 & -3 & 7 & 8 \end{array}\right] \]

  2. \[ x_{1} \left[\begin{array}{c} 6 \\ -5 \\ 5 \\ -5 \end{array}\right] + x_{2} \left[\begin{array}{c} 5 \\ -4 \\ 3 \\ -3 \end{array}\right] + x_{3} \left[\begin{array}{c} -7 \\ 6 \\ -6 \\ 7 \end{array}\right] = \left[\begin{array}{c} -4 \\ 4 \\ -7 \\ 8 \end{array}\right] \]


Example 17 πŸ”—

Consider the vector equation.

\[ x_{1} \left[\begin{array}{c} 1 \\ 0 \\ 0 \\ -1 \end{array}\right] + x_{2} \left[\begin{array}{c} 2 \\ 0 \\ 0 \\ -2 \end{array}\right] + x_{3} \left[\begin{array}{c} 5 \\ 1 \\ 1 \\ -1 \end{array}\right] = \left[\begin{array}{c} 8 \\ 2 \\ 2 \\ 0 \end{array}\right] \]

  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{alignat*}{4} x &+& 2 \, y &+& 5 \, z &=& 8 \\ & & & & z &=& 2 \\ & & & & z &=& 2 \\-x &-& 2 \, y &-& z &=& 0 \\ \end{alignat*}
  2. \[ \left[\begin{array}{ccc|c} 1 & 2 & 5 & 8 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 1 & 2 \\ -1 & -2 & -1 & 0 \end{array}\right] \]


Example 18 πŸ”—

Consider the augmented matrix

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

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


Example 19 πŸ”—

Consider the system of equations \begin{alignat*}{4} x &+& 5 \, y & & &=& 0 \\-x &-& 4 \, y & & &=& 0 \\ &-& y &+& z &=& 3 \\-2 \, x &-& 5 \, y &-& 2 \, z &=& -6 \\ \end{alignat*}

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

Answer:

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

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


Example 20 πŸ”—

Consider the augmented matrix

\[ \left[\begin{array}{ccc|c} -7 & -5 & 1 & -5 \\ -4 & -3 & 0 & -2 \\ -2 & -1 & 3 & -6 \\ 5 & 3 & 0 & 1 \end{array}\right] \]

  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{alignat*}{4} -7 \, x &-& 5 \, y &+& z &=& -5 \\-4 \, x &-& 3 \, y & & &=& -2 \\-2 \, x &-& y &+& 3 \, z &=& -6 \\5 \, x &+& 3 \, y & & &=& 1 \\ \end{alignat*}
  2. \[ x_{1} \left[\begin{array}{c} -7 \\ -4 \\ -2 \\ 5 \end{array}\right] + x_{2} \left[\begin{array}{c} -5 \\ -3 \\ -1 \\ 3 \end{array}\right] + x_{3} \left[\begin{array}{c} 1 \\ 0 \\ 3 \\ 0 \end{array}\right] = \left[\begin{array}{c} -5 \\ -2 \\ -6 \\ 1 \end{array}\right] \]


Example 21 πŸ”—

Consider the vector equation.

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

  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{alignat*}{5} x &+& 2 \, y &+& 3 \, z &+& 4 \, {w} &=& -3 \\ & & y &+& z &+& {w} &=& -1 \\ & & 2 \, y &+& 3 \, z &+& 4 \, {w} &=& -4 \\ \end{alignat*}
  2. \[ \left[\begin{array}{cccc|c} 1 & 2 & 3 & 4 & -3 \\ 0 & 1 & 1 & 1 & -1 \\ 0 & 2 & 3 & 4 & -4 \end{array}\right] \]


Example 22 πŸ”—

Consider the augmented matrix

\[ \left[\begin{array}{ccc|c} 4 & -6 & -7 & 0 \\ 1 & -2 & -1 & -3 \\ -1 & 1 & 2 & -2 \\ -2 & 5 & 3 & 7 \end{array}\right] \]

  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{alignat*}{4} 4 \, x &-& 6 \, y &-& 7 \, z &=& 0 \\x &-& 2 \, y &-& z &=& -3 \\-x &+& y &+& 2 \, z &=& -2 \\-2 \, x &+& 5 \, y &+& 3 \, z &=& 7 \\ \end{alignat*}
  2. \[ x_{1} \left[\begin{array}{c} 4 \\ 1 \\ -1 \\ -2 \end{array}\right] + x_{2} \left[\begin{array}{c} -6 \\ -2 \\ 1 \\ 5 \end{array}\right] + x_{3} \left[\begin{array}{c} -7 \\ -1 \\ 2 \\ 3 \end{array}\right] = \left[\begin{array}{c} 0 \\ -3 \\ -2 \\ 7 \end{array}\right] \]


Example 23 πŸ”—

Consider the system of equations \begin{alignat*}{5} -x &-& 2 \, y &+& 5 \, z &-& 3 \, {w} &=& -1 \\2 \, x &+& 3 \, y &-& 8 \, z &+& 4 \, {w} &=& 2 \\4 \, x &+& y &-& 6 \, z &-& 2 \, {w} &=& 4 \\ \end{alignat*}

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

Answer:

  1. \[ \left[\begin{array}{cccc|c} -1 & -2 & 5 & -3 & -1 \\ 2 & 3 & -8 & 4 & 2 \\ 4 & 1 & -6 & -2 & 4 \end{array}\right] \]

  2. \[ x_{1} \left[\begin{array}{c} -1 \\ 2 \\ 4 \end{array}\right] + x_{2} \left[\begin{array}{c} -2 \\ 3 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} 5 \\ -8 \\ -6 \end{array}\right] + x_{4} \left[\begin{array}{c} -3 \\ 4 \\ -2 \end{array}\right] = \left[\begin{array}{c} -1 \\ 2 \\ 4 \end{array}\right] \]


Example 24 πŸ”—

Consider the augmented matrix

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

  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{alignat*}{5} x &-& 2 \, y &+& z &+& {w} &=& 1 \\ & & & & z &+& 2 \, {w} &=& 1 \\x &-& 2 \, y &+& 4 \, z &+& 7 \, {w} &=& 4 \\ \end{alignat*}
  2. \[ x_{1} \left[\begin{array}{c} 1 \\ 0 \\ 1 \end{array}\right] + x_{2} \left[\begin{array}{c} -2 \\ 0 \\ -2 \end{array}\right] + x_{3} \left[\begin{array}{c} 1 \\ 1 \\ 4 \end{array}\right] + x_{4} \left[\begin{array}{c} 1 \\ 2 \\ 7 \end{array}\right] = \left[\begin{array}{c} 1 \\ 1 \\ 4 \end{array}\right] \]


Example 25 πŸ”—

Consider the vector equation.

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

  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{alignat*}{4} x_{1} & & &-& 2 \, x_{3} &=& 2 \\ & & x_{2} & & &=& 2 \\-x_{1} & & &+& 2 \, x_{3} &=& -2 \\x_{1} &+& x_{2} &-& 2 \, x_{3} &=& 4 \\ \end{alignat*}
  2. \[ \left[\begin{array}{ccc|c} 1 & 0 & -2 & 2 \\ 0 & 1 & 0 & 2 \\ -1 & 0 & 2 & -2 \\ 1 & 1 & -2 & 4 \end{array}\right] \]


Example 26 πŸ”—

Consider the system of equations \begin{alignat*}{5} 2 \, x_{1} &-& 4 \, x_{2} &-& 5 \, x_{3} &+& 4 \, x_{4} &=& -5 \\-x_{1} &+& 2 \, x_{2} &+& 3 \, x_{3} &-& 2 \, x_{4} &=& 3 \\x_{1} &-& 3 \, x_{2} &-& 6 \, x_{3} &+& 2 \, x_{4} &=& -6 \\ \end{alignat*}

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

Answer:

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

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


Example 27 πŸ”—

Consider the augmented matrix

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

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


Example 28 πŸ”—

Consider the system of equations \begin{alignat*}{5} &-& x_{2} &+& 3 \, x_{3} &+& 6 \, x_{4} &=& -3 \\x_{1} & & &+& 4 \, x_{3} &+& 3 \, x_{4} &=& 1 \\-x_{1} & & &-& 4 \, x_{3} &-& 2 \, x_{4} &=& -1 \\ \end{alignat*}

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

Answer:

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

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


Example 29 πŸ”—

Consider the augmented matrix

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

  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{alignat*}{4} -2 \, x &+& y &+& 6 \, z &=& 0 \\x &-& y &-& 4 \, z &=& 0 \\x &-& y &-& 4 \, z &=& 0 \\ &-& 3 \, y &-& 6 \, z &=& 0 \\ \end{alignat*}
  2. \[ x_{1} \left[\begin{array}{c} -2 \\ 1 \\ 1 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} 1 \\ -1 \\ -1 \\ -3 \end{array}\right] + x_{3} \left[\begin{array}{c} 6 \\ -4 \\ -4 \\ -6 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \]


Example 30 πŸ”—

Consider the vector equation.

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

  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{alignat*}{5} x_{1} &-& 4 \, x_{2} &+& 3 \, x_{3} & & &=& -2 \\ & & & & & & x_{4} &=& 3 \\-x_{1} &+& 4 \, x_{2} &-& 3 \, x_{3} &-& 2 \, x_{4} &=& -4 \\ \end{alignat*}
  2. \[ \left[\begin{array}{cccc|c} 1 & -4 & 3 & 0 & -2 \\ 0 & 0 & 0 & 1 & 3 \\ -1 & 4 & -3 & -2 & -4 \end{array}\right] \]


Example 31 πŸ”—

Consider the vector equation.

\[ x_{1} \left[\begin{array}{c} 1 \\ -1 \\ 0 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} 2 \\ -3 \\ 3 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} 0 \\ 4 \\ -5 \\ -3 \end{array}\right] = \left[\begin{array}{c} 7 \\ 2 \\ -6 \\ -6 \end{array}\right] \]

  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{alignat*}{4} x &+& 2 \, y & & &=& 7 \\-x &-& 3 \, y &+& 4 \, z &=& 2 \\ & & 3 \, y &-& 5 \, z &=& -6 \\ & & y &-& 3 \, z &=& -6 \\ \end{alignat*}
  2. \[ \left[\begin{array}{ccc|c} 1 & 2 & 0 & 7 \\ -1 & -3 & 4 & 2 \\ 0 & 3 & -5 & -6 \\ 0 & 1 & -3 & -6 \end{array}\right] \]


Example 32 πŸ”—

Consider the augmented matrix

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

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


Example 33 πŸ”—

Consider the system of equations \begin{alignat*}{4} &-& x_{2} & & &=& 0 \\ & & x_{2} &+& 2 \, x_{3} &=& -6 \\x_{1} &+& x_{2} &+& x_{3} &=& -4 \\-x_{1} &-& 3 \, x_{2} &-& 2 \, x_{3} &=& 7 \\ \end{alignat*}

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

Answer:

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

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


Example 34 πŸ”—

Consider the system of equations \begin{alignat*}{5} x_{1} &-& x_{2} &-& 6 \, x_{3} &-& 8 \, x_{4} &=& -3 \\ & & x_{2} &+& 3 \, x_{3} &+& 4 \, x_{4} &=& 3 \\ & & & & x_{3} &+& x_{4} &=& 1 \\ \end{alignat*}

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

Answer:

  1. \[ \left[\begin{array}{cccc|c} 1 & -1 & -6 & -8 & -3 \\ 0 & 1 & 3 & 4 & 3 \\ 0 & 0 & 1 & 1 & 1 \end{array}\right] \]

  2. \[ x_{1} \left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} -1 \\ 1 \\ 0 \end{array}\right] + x_{3} \left[\begin{array}{c} -6 \\ 3 \\ 1 \end{array}\right] + x_{4} \left[\begin{array}{c} -8 \\ 4 \\ 1 \end{array}\right] = \left[\begin{array}{c} -3 \\ 3 \\ 1 \end{array}\right] \]


Example 35 πŸ”—

Consider the augmented matrix

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

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


Example 36 πŸ”—

Consider the augmented matrix

\[ \left[\begin{array}{ccc|c} -1 & 0 & 5 & -6 \\ -2 & -7 & -8 & 6 \\ -1 & -2 & 0 & -1 \\ 1 & 3 & 8 & -7 \end{array}\right] \]

  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{alignat*}{4} -x_{1} & & &+& 5 \, x_{3} &=& -6 \\-2 \, x_{1} &-& 7 \, x_{2} &-& 8 \, x_{3} &=& 6 \\-x_{1} &-& 2 \, x_{2} & & &=& -1 \\x_{1} &+& 3 \, x_{2} &+& 8 \, x_{3} &=& -7 \\ \end{alignat*}
  2. \[ x_{1} \left[\begin{array}{c} -1 \\ -2 \\ -1 \\ 1 \end{array}\right] + x_{2} \left[\begin{array}{c} 0 \\ -7 \\ -2 \\ 3 \end{array}\right] + x_{3} \left[\begin{array}{c} 5 \\ -8 \\ 0 \\ 8 \end{array}\right] = \left[\begin{array}{c} -6 \\ 6 \\ -1 \\ -7 \end{array}\right] \]


Example 37 πŸ”—

Consider the augmented matrix

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

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


Example 38 πŸ”—

Consider the system of equations \begin{alignat*}{5} -x &-& y &-& 2 \, z &+& 3 \, {w} &=& 3 \\-x &+& 2 \, y &-& z &-& 5 \, {w} &=& 6 \\2 \, x & & &+& 3 \, z &-& {w} &=& -7 \\ \end{alignat*}

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

Answer:

  1. \[ \left[\begin{array}{cccc|c} -1 & -1 & -2 & 3 & 3 \\ -1 & 2 & -1 & -5 & 6 \\ 2 & 0 & 3 & -1 & -7 \end{array}\right] \]

  2. \[ x_{1} \left[\begin{array}{c} -1 \\ -1 \\ 2 \end{array}\right] + x_{2} \left[\begin{array}{c} -1 \\ 2 \\ 0 \end{array}\right] + x_{3} \left[\begin{array}{c} -2 \\ -1 \\ 3 \end{array}\right] + x_{4} \left[\begin{array}{c} 3 \\ -5 \\ -1 \end{array}\right] = \left[\begin{array}{c} 3 \\ 6 \\ -7 \end{array}\right] \]


Example 39 πŸ”—

Consider the vector equation.

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

  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{alignat*}{5} 2 \, x_{1} &-& 6 \, x_{2} &+& x_{3} & & &=& -6 \\x_{1} & & &+& 3 \, x_{3} &+& 5 \, x_{4} &=& -5 \\x_{1} &-& 5 \, x_{2} &-& x_{3} &-& 3 \, x_{4} &=& -2 \\ \end{alignat*}
  2. \[ \left[\begin{array}{cccc|c} 2 & -6 & 1 & 0 & -6 \\ 1 & 0 & 3 & 5 & -5 \\ 1 & -5 & -1 & -3 & -2 \end{array}\right] \]


Example 40 πŸ”—

Consider the vector equation.

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

  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{alignat*}{5} x_{1} &+& 3 \, x_{2} &+& 3 \, x_{3} & & &=& -3 \\-3 \, x_{1} &-& 5 \, x_{2} &-& 6 \, x_{3} &+& x_{4} &=& 3 \\ & & x_{2} &+& x_{3} & & &=& -1 \\ \end{alignat*}
  2. \[ \left[\begin{array}{cccc|c} 1 & 3 & 3 & 0 & -3 \\ -3 & -5 & -6 & 1 & 3 \\ 0 & 1 & 1 & 0 & -1 \end{array}\right] \]


Example 41 πŸ”—

Consider the system of equations \begin{alignat*}{5} x_{1} &+& 5 \, x_{2} &-& x_{3} &+& 3 \, x_{4} &=& 0 \\x_{1} &+& 5 \, x_{2} & & &+& x_{4} &=& 2 \\x_{1} &+& 5 \, x_{2} &-& x_{3} &+& 3 \, x_{4} &=& 0 \\ \end{alignat*}

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

Answer:

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

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


Example 42 πŸ”—

Consider the augmented matrix

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

  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{alignat*}{5} x &-& y & & &+& 2 \, {w} &=& 6 \\ & & y &-& 2 \, z &-& {w} &=& -4 \\ & & y &-& 2 \, z & & &=& -3 \\ \end{alignat*}
  2. \[ x_{1} \left[\begin{array}{c} 1 \\ 0 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} -1 \\ 1 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} 0 \\ -2 \\ -2 \end{array}\right] + x_{4} \left[\begin{array}{c} 2 \\ -1 \\ 0 \end{array}\right] = \left[\begin{array}{c} 6 \\ -4 \\ -3 \end{array}\right] \]


Example 43 πŸ”—

Consider the system of equations \begin{alignat*}{4} -x &+& y &-& z &=& -4 \\-x & & &-& 4 \, z &=& -5 \\x &-& y &+& 2 \, z &=& 5 \\-x & & &-& 7 \, z &=& -8 \\ \end{alignat*}

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

Answer:

  1. \[ \left[\begin{array}{ccc|c} -1 & 1 & -1 & -4 \\ -1 & 0 & -4 & -5 \\ 1 & -1 & 2 & 5 \\ -1 & 0 & -7 & -8 \end{array}\right] \]

  2. \[ x_{1} \left[\begin{array}{c} -1 \\ -1 \\ 1 \\ -1 \end{array}\right] + x_{2} \left[\begin{array}{c} 1 \\ 0 \\ -1 \\ 0 \end{array}\right] + x_{3} \left[\begin{array}{c} -1 \\ -4 \\ 2 \\ -7 \end{array}\right] = \left[\begin{array}{c} -4 \\ -5 \\ 5 \\ -8 \end{array}\right] \]


Example 44 πŸ”—

Consider the augmented matrix

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

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


Example 45 πŸ”—

Consider the vector equation.

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

  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{alignat*}{4} x_{1} &+& 5 \, x_{2} &-& 3 \, x_{3} &=& 3 \\ & & & & x_{3} &=& -1 \\-x_{1} &-& 5 \, x_{2} &+& 4 \, x_{3} &=& -4 \\x_{1} &+& 5 \, x_{2} & & &=& 0 \\ \end{alignat*}
  2. \[ \left[\begin{array}{ccc|c} 1 & 5 & -3 & 3 \\ 0 & 0 & 1 & -1 \\ -1 & -5 & 4 & -4 \\ 1 & 5 & 0 & 0 \end{array}\right] \]


Example 46 πŸ”—

Consider the system of equations \begin{alignat*}{5} x & & &-& 2 \, z &+& {w} &=& -8 \\x &+& y &-& 2 \, z &+& 3 \, {w} &=& -6 \\ &-& 4 \, y &+& z &-& 8 \, {w} &=& -4 \\ \end{alignat*}

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

Answer:

  1. \[ \left[\begin{array}{cccc|c} 1 & 0 & -2 & 1 & -8 \\ 1 & 1 & -2 & 3 & -6 \\ 0 & -4 & 1 & -8 & -4 \end{array}\right] \]

  2. \[ x_{1} \left[\begin{array}{c} 1 \\ 1 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} 0 \\ 1 \\ -4 \end{array}\right] + x_{3} \left[\begin{array}{c} -2 \\ -2 \\ 1 \end{array}\right] + x_{4} \left[\begin{array}{c} 1 \\ 3 \\ -8 \end{array}\right] = \left[\begin{array}{c} -8 \\ -6 \\ -4 \end{array}\right] \]


Example 47 πŸ”—

Consider the system of equations \begin{alignat*}{5} -x &+& y &+& 5 \, z &+& 3 \, {w} &=& 8 \\-x & & &+& z & & &=& 2 \\-3 \, x &-& 2 \, y &-& 4 \, z &-& 5 \, {w} &=& -5 \\ \end{alignat*}

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

Answer:

  1. \[ \left[\begin{array}{cccc|c} -1 & 1 & 5 & 3 & 8 \\ -1 & 0 & 1 & 0 & 2 \\ -3 & -2 & -4 & -5 & -5 \end{array}\right] \]

  2. \[ x_{1} \left[\begin{array}{c} -1 \\ -1 \\ -3 \end{array}\right] + x_{2} \left[\begin{array}{c} 1 \\ 0 \\ -2 \end{array}\right] + x_{3} \left[\begin{array}{c} 5 \\ 1 \\ -4 \end{array}\right] + x_{4} \left[\begin{array}{c} 3 \\ 0 \\ -5 \end{array}\right] = \left[\begin{array}{c} 8 \\ 2 \\ -5 \end{array}\right] \]


Example 48 πŸ”—

Consider the vector equation.

\[ x_{1} \left[\begin{array}{c} 1 \\ 0 \\ -2 \end{array}\right] + x_{2} \left[\begin{array}{c} -1 \\ 1 \\ 0 \end{array}\right] + x_{3} \left[\begin{array}{c} 1 \\ 3 \\ -8 \end{array}\right] + x_{4} \left[\begin{array}{c} 6 \\ -3 \\ -5 \end{array}\right] = \left[\begin{array}{c} 1 \\ 0 \\ -2 \end{array}\right] \]

  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{alignat*}{5} x_{1} &-& x_{2} &+& x_{3} &+& 6 \, x_{4} &=& 1 \\ & & x_{2} &+& 3 \, x_{3} &-& 3 \, x_{4} &=& 0 \\-2 \, x_{1} & & &-& 8 \, x_{3} &-& 5 \, x_{4} &=& -2 \\ \end{alignat*}
  2. \[ \left[\begin{array}{cccc|c} 1 & -1 & 1 & 6 & 1 \\ 0 & 1 & 3 & -3 & 0 \\ -2 & 0 & -8 & -5 & -2 \end{array}\right] \]


Example 49 πŸ”—

Consider the vector equation.

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

  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{alignat*}{4} & & 2 \, x_{2} &+& 2 \, x_{3} &=& 4 \\-x_{1} &-& 3 \, x_{2} &-& 4 \, x_{3} &=& -6 \\ &-& x_{2} & & &=& -1 \\-x_{1} &-& 2 \, x_{2} &-& 3 \, x_{3} &=& -4 \\ \end{alignat*}
  2. \[ \left[\begin{array}{ccc|c} 0 & 2 & 2 & 4 \\ -1 & -3 & -4 & -6 \\ 0 & -1 & 0 & -1 \\ -1 & -2 & -3 & -4 \end{array}\right] \]


Example 50 πŸ”—

Consider the augmented matrix

\[ \left[\begin{array}{cccc|c} 1 & 2 & 3 & -4 & 8 \\ -1 & 0 & 4 & -5 & 3 \\ 0 & -1 & -3 & 4 & -5 \end{array}\right] \]

  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{alignat*}{5} x &+& 2 \, y &+& 3 \, z &-& 4 \, {w} &=& 8 \\-x & & &+& 4 \, z &-& 5 \, {w} &=& 3 \\ &-& y &-& 3 \, z &+& 4 \, {w} &=& -5 \\ \end{alignat*}
  2. \[ x_{1} \left[\begin{array}{c} 1 \\ -1 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} 2 \\ 0 \\ -1 \end{array}\right] + x_{3} \left[\begin{array}{c} 3 \\ 4 \\ -3 \end{array}\right] + x_{4} \left[\begin{array}{c} -4 \\ -5 \\ 4 \end{array}\right] = \left[\begin{array}{c} 8 \\ 3 \\ -5 \end{array}\right] \]