LearnViz CompSci
Home / Learn / Math for Computing / Gaussian Elimination

Intermediate Topic 27 of 51 Math for Computing

Gaussian Elimination

Row operations solving linear systems.

How it works

Gaussian elimination solves linear systems by using row operations, swapping rows, scaling rows and adding multiples of one row to another, to reduce the augmented matrix to row echelon form, then solving by back substitution. Gauss-Jordan elimination continues to reduced row echelon form, giving the solution directly. Partial pivoting selects the largest available entry as pivot to limit numerical error. The cost is O(n cubed), and the process also computes rank, determinants and inverses, and is equivalent to LU factorization.

Where it is used

Solving small to medium dense systems in engineering and science, computing inverses, finding bases and checking linear independence use Gaussian elimination.

Common pitfall

Without pivoting, a zero or tiny pivot causes division by zero or large errors. Floating-point rounding accumulates for ill-conditioned matrices. Fill-in destroys sparsity for large sparse problems, where iterative or sparse direct solvers are better. Row operations must be applied to the right-hand side consistently. Keep track of row swaps when computing determinants.

Where it fits in the course

Gaussian Elimination is topic 27 of 51 in Math for Computing, rated Intermediate. It follows Linear Systems. The next topic is Eigenvalues and Eigenvectors.

Practice and explore

How to study this topic

  1. Read the explanation above and restate the definition in your own words.
  2. Open the interactive lesson and answer its mini quiz.
  3. Read the common pitfall, then write a short example that avoids it.
  4. Check your understanding in the Math for Computing quiz, and revisit the topic if you miss a question.

Frequently asked questions

What is Gaussian Elimination?
Row operations solving linear systems. It belongs to Math for Computing and is an intermediate-level topic.
Where do people use Gaussian Elimination?
Solving small to medium dense systems in engineering and science, computing inverses, finding bases and checking linear independence use Gaussian elimination.
What is the most common mistake with Gaussian Elimination?
Without pivoting, a zero or tiny pivot causes division by zero or large errors.
What should I learn before this topic?
Start with Linear Systems, the topic just before it in Math for Computing.
How can I practice it?
Answer the mini quiz in the interactive lesson, then take the Math for Computing quiz.
Is Gaussian Elimination hard to learn?
It is an Intermediate topic. Be comfortable with the Beginner topics in this area first, then read the common pitfall carefully, since it is where most mistakes happen.
How long does it take to learn Gaussian Elimination?
Most learners can read the explanation and finish the interactive lesson in under an hour. Fluency comes from applying the idea, so plan a few short practice sessions with the quiz.
Which topics are related to Gaussian Elimination?
Linear Systems, Eigenvalues and Eigenvectors, Confidence Intervals are closely connected. The full list is in the related topics section of this page.
What should I study after Gaussian Elimination?
Continue with Eigenvalues and Eigenvectors, the next topic in Math for Computing.

Related topics