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.

Glossary

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

row-equivalent

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

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