Summary: Gaussian Elimination

Key Concepts

  • An augmented matrix is one that contains the coefficients and constants of a system of equations.
  • A matrix augmented with the constant column can be represented as the original system of equations.
  • Row operations include multiplying a row by a constant, adding one row to another row, and interchanging rows.
  • We can use Gaussian elimination to solve a system of equations.
  • Row operations are performed on matrices to obtain row-echelon form.
  • To solve a system of equations, write it in augmented matrix form. Perform row operations to obtain row-echelon form. Back-substitute to find the solutions.
  • A calculator can be used to solve systems of equations using matrices.
  • Many real-world problems can be solved using augmented matrices.


augmented matrix
a coefficient matrix adjoined with the constant column separated by a vertical line within the matrix brackets
coefficient matrix
a matrix that contains only the coefficients from a system of equations
Gaussian elimination
using elementary row operations to obtain a matrix in row-echelon form
main diagonal
entries from the upper left corner diagonally to the lower right corner of a square matrix
row-echelon form
after performing row operations, the matrix form that contains ones down the main diagonal and zeros at every space below the diagonal
two matrices [latex]A[/latex] and [latex]B[/latex] are row-equivalent if one can be obtained from the other by performing basic row operations
row operations
adding one row to another row, multiplying a row by a constant, interchanging rows, and so on, with the goal of achieving row-echelon form