Matrix trainer: practise Gaussian elimination
You do the work: enter row operations like II − 3·I and the matrix changes at once. Three tasks, new every day, with hints and worked solutions. Free and without an account.
Keep practising for free, with your progress
Systems up to 3×3, determinant and rank without a daily limit in a free account. From €4.99 once or with Plus, also 4×4, inverses and solvability.
Solve the system of linear equations.
Use row operations until you can read off the solution, then enter it below.
| −1 | −4 | 7 |
| −1 | −2 | 1 |
x₁, x₂ | b
Your answer
Topics with explanations and practiceGaussian elimination · Determinant · Rank of a matrix …
Gaussian elimination
In Gaussian elimination you bring the augmented matrix to row echelon form with row operations and then substitute back from the bottom up. Here you do the work: you enter each row operation and the matrix changes at once.
Determinant
For 3×3 and 4×4 matrices Gaussian elimination is often the fastest way: bring the matrix to triangular form, multiply the diagonal and keep track of what the row operations did to the determinant.
Rank of a matrix
The rank is the number of linearly independent rows. The easiest way to find it is Gaussian elimination: reach row echelon form and count the rows that are not all zeros.
Inverse matrix
You find the inverse with Gauss–Jordan elimination: write the identity matrix next to A and keep applying row operations until the identity is on the left. A⁻¹ is then on the right.
Solvability of linear systems
A linear system has either exactly one, no or infinitely many solutions. Which case applies shows up in the row echelon form of the augmented matrix.
How the matrix trainer works
Do it yourself, do not just watch
You enter every row operation yourself, just like on paper. The trainer calculates exactly with fractions and warns you when a zero you created gets lost.
A hint when you are stuck
"Next step" suggests a sensible operation for your current matrix, not just for the starting one.
Tasks with clean results
Every task is built backwards from a whole-number answer. No awkward numbers like 17/23, and the answer is always right.