Elementary Row Operations

  • (Interchange) Interchange two equations
  • (Scaling) Multiply an equation by a non-zero constant
  • (Replacement) Replace one row by the sum of itself and a multiple of another row [like r2'= -3r1 + r2]

    Systematic Method to achieve Gaussian Elimination

  • Save the x term in equation 1 and use it to eliminate all the other x terms below it via rk' = c r1 + rk
  • Ignore equation 1 and use the y term in equation 2 to eliminate all the y terms below it.
  • Continue until the matrix is in Gaussian or echelon form, with 0s below the diagonal (interchange rows as needed)
  • If the system is consistent then the last row with non-zero coeffients will yield xk=b, and then back substitution can be used to solve for the variables.

    Continuing to Gauss-Jordan/ReducedRowEchelon form

  • Scale the last row with non-zero coefficients so that the diagonal entry is a 1.
  • Use the last non-zero equation to eliminate the spots above it
  • Repeate these steps using the second last non-zero equation.
  • Continue until the matrix is in Gauss-Jordan/ReducedRowEchelon form with 0s or 1s on the diagonal and 0 coeffients everywhere else.
  • Read off the solutions