Gaussian elimination

Definition

Gaussian elimination is a method for solving systems of linear equations by transforming the system's augmented matrix into row-echelon form using elementary row operations. This process allows for easy back-substitution to find the solutions.

5 Must Know Facts For Your Next Test

1. Gaussian elimination involves three types of elementary row operations: swapping rows, multiplying a row by a non-zero scalar, and adding or subtracting multiples of rows.
2. The goal of Gaussian elimination is to transform the augmented matrix into an upper triangular form (row-echelon form).
3. In row-echelon form, all entries below the main diagonal are zeros.
4. If a system has no solutions, it will be indicated by a row in the augmented matrix where all coefficients are zero but the constant term is non-zero.
5. After achieving row-echelon form, back-substitution starts from the last equation and works upwards to find the values of variables.

Review Questions

• What are the three types of elementary row operations used in Gaussian elimination?
• What does it mean for an augmented matrix to be in row-echelon form?
• How can you identify if a system of equations has no solution using Gaussian elimination?

Related terms

Row-Echelon Form: A matrix is in row-echelon form when all non-zero rows are above any rows of all zeros, and each leading entry (the first non-zero number from the left) of a row is in a column to the right of the leading entry of the previous row.

Back-Substitution: The process used after Gaussian elimination to solve for variables starting from the last equation and working upwards.

Elementary Row Operations: Operations that include swapping two rows, multiplying a row by a non-zero scalar, and adding or subtracting multiples of one row to another.