Gaussian elimination

Definition

Gaussian elimination is a method for solving systems of linear equations. It transforms the system's augmented matrix into row-echelon form using row operations.

5 Must Know Facts For Your Next Test

1. Gaussian elimination involves three types of row operations: swapping rows, multiplying a row by a nonzero scalar, and adding or subtracting rows.
2. The goal is to convert the augmented matrix into an upper triangular form where all elements below the main diagonal are zero.
3. Once in row-echelon form, back-substitution is used to find the solutions to the system of equations.
4. If an equation becomes $0 = c$ (where $c$ is a non-zero constant), the system has no solution and is considered inconsistent.
5. If an entire row of zeros appears in the coefficient matrix, it indicates infinitely many solutions if consistent.

Review Questions

• What are the three types of row operations used in Gaussian elimination?
• How can you determine if a system has no solution during Gaussian elimination?
• What does an entire row of zeros in the coefficient matrix signify about the system's solutions?

Related terms

Row-Echelon Form: A matrix form where all nonzero rows are above any rows of all zeros, and each leading coefficient (first nonzero number from left) is to the right of the leading coefficient in the previous row.

Back-Substitution:

A method used after Gaussian elimination where you solve for variables starting from the last equation up to the first.

Augmented Matrix:

A matrix that includes both the coefficients and constants from a system of linear equations.