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Augmented Matrix

  • Let \(Ax = b\) be a system of linear equations where \(A\) is an \(m \times n\) matrix and \(b\) is an \(m \times 1\) matrix.
  • The augmented matrix of \(A\) and \(b\) is the \(m \times (n+1)\) matrix \([A|b]\).

Example

Lets say for a system of linear equations:

\(\begin{alignedat}{4} 3x_1 & {}+{} & 2x_2 & {}+{} & x_3 & {}+{} & x_4 &= 6 \\ x_1 & {}+{} & x_2 & & & & &= 2 \\ & & 7x_2 & {}+{} & x_3 & {}+{} & x_4 &= 8 \end{alignedat}\)

  • Matrix \(A = \begin{bmatrix} 3 & 2 & 1 & 1 \\ 1 & 1 & 0 & 0 \\ 0 & 7 & 1 & 1 \\ \end{bmatrix}\)

  • Matrix \(b = \begin{bmatrix} 6 \\ 2 \\ 8 \\ \end{bmatrix}\)

  • Augmented matrix \(= [A|b] = \left[ \begin{array}{cccc|c} 3 & 2 & 1 & 1 & 6 \\ 1 & 1 & 0 & 0 & 2 \\ 0 & 7 & 1 & 1& 8 \\ \end{array} \right]\)

Gaussian Elimination

Gaussian elimination is a method for solving a system of linear equations.

Algorithm

  • Form the augmented matrix \([A|b]\).
  • Perform the following steps until the augmented matrix is in reduced row echelon form.
  • Apply elementary row operations on both sides of the augmented matrix.
  • It is ok if the \(b\) column is not in reduced row echelon form.
  • Let \(R\) be the submatrix of the obtained matrix of the frist \(n\) columns and \(c\) be the submatrix of the obtained matrix consisting of the last column.
  • We write the obtained matrix as \([R|c]\).
  • The solutions of \(Ax=b\) are precisely the values of solutions of \(Rx=c\).

Note

If there are zero rows in the augmented matrix, then the system has no solutions.

Example

  • \(A = \left[\begin{array}{cccc|c} 3 & 2 & 1 & 1 & 6 \\ 1 & 1 & 0 & 0 & 2 \\ 0 & 7 & 1 & 1& 8 \\ \end{array} \right]\)
  • \(A\) is not in reduced row echelon form.
  • \(R_1/3\):
    • \(A = \left[ \begin{array}{cccc|c} 1 & \frac{2}{3} & \frac{1}{3} & \frac{1}{3} & 2 \\ 1 & 1 & 0 & 0 & 2 \\ 0 & 7 & 1 & 1 & 8 \\ \end{array} \right]\)

  • \(R_2 - R_1\):

    • \(A = \left[\begin{array}{cccc|c} 1 & \frac{2}{3} & \frac{1}{3} & \frac{1}{3} & 2 \\ 0 & \frac{1}{3} & -\frac{1}{3} & -\frac{1}{3} & 0 \\ 0 & 7 & 1 & 1& 8 \\ \end{array}\right]\)

  • \(3R_2\):

    • \(A = \left[\begin{array}{cccc|c} 1 & \frac{2}{3} & \frac{1}{3} & \frac{1}{3} & 2 \\ 0 & 1 & -1 & -1 & 0 \\ 0 & 7 & 1 & 1& 8 \\ \end{array} \right]\)
  • \(R_3 - 7R_2\):
    • \(A = \left[\begin{array}{cccc|c} 1 & \frac{2}{3} & \frac{1}{3} & \frac{1}{3} & 2 \\ 0 & 1 & -1 & -1 & 0 \\ 0 & 0 & 8 & 8 & 8 \\ \end{array} \right]\)
  • \(R_3/8\):
    • \(A = \left[\begin{array}{cccc|c} 1 & \frac{2}{3} & \frac{1}{3} & \frac{1}{3} & 2 \\ 0 & 1 & -1 & -1 & 0 \\ 0 & 0 & 1 & 1 & 1 \\ \end{array}\right]\)
  • \(R_2 + R_3, R_1 - R_3/3\)
    • \(A = \left[\begin{array}{cccc|c} 1 & \frac{2}{3} & 0 & 0 & \frac{5}{3} \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 1 & 1 \\ \end{array}\right]\)
  • \(R_1 - 2/3R_2:\)
    • \(A = \left[\begin{array}{cccc|c} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 1 & 1 \\ \end{array} \right]\)

Homogeneous System of linear equations

  • 0 is always a solution of a homogenous system of linear equations.
  • \(Ax = 0\) is called a trivial solution.
  • For a homogenous system, there are always \(2\) types of possible outcomes:
  • 0 is the unqiue solution.
  • There are infinitely many solutions other than 0.
  • In a homogenous system, if there are more variables than equations, then it is guaranteed to have non-trivial solutions.