Solve a System of Linear Equations with Matrices
Write the system as A·x = b, where A holds the coefficients and b the constants; if det(A) ≠ 0 the unique solution is x = A⁻¹·b.
At a glance
| What it is | Solve systems of linear equations with matrices: write Ax = b and compute x = A⁻¹b. Exact fraction answers, 2 to 10 unknowns. Free. |
|---|---|
| Type | Interactive guide with a calculator |
| Level | High school and college |
| Price | Free, with no account or sign-up |
| Works offline | Yes, after the first visit; installs as an app |
| Made by | Aalok (aalok.builds@gmail.com) |
| Last updated | 1 November 2026 |
Key facts
| Matrix form | A·x = b |
|---|---|
| Solution | x = A⁻¹·b when det(A) ≠ 0 |
| Example | 2x + y = 5 and x + 3y = 10 give x = 1, y = 3 |
| No unique solution | If det(A) = 0, use rref on the augmented matrix [A | b] |
How do you set up the matrices?
For 2x + y = 5 and x + 3y = 10, A = [[2, 1], [1, 3]] holds the coefficients and B = [[5], [10]] the right-hand sides. inv(A)*B returns [[1], [3]], so x = 1 and y = 3.
How to use it in Plotline
- A (coefficients) and B (constants, one column) are filled in, and inv(A)*B is solved.
- Change the numbers for your own system; add rows and columns for more unknowns.
- If the answer says the matrix is singular, try rref on the augmented matrix instead.
Frequently asked questions
Can it solve 3 equations with 3 unknowns?
Yes. Enter the 3×3 coefficient matrix as A and the three constants as a 3×1 matrix B, then compute inv(A)·B. The same method works for larger systems, as long as A is square and its determinant is not 0.
Is it free?
Yes. The matrix calculator is free with no account and no sign-up, and there is no limit on how many matrices you work with. Determinants, inverses, RREF, multiplication and systems of equations are all included, and it works offline once loaded.
Learn more
Related
- Inverse matrix calculator – Find the inverse of a 2×2, 3×3 or larger matrix online with exact fraction answers.
- RREF calculator – Free RREF calculator: reduce any matrix to reduced row echelon form with exact fractions.
- Determinant calculator – Free determinant calculator for 2×2, 3×3 and larger matrices up to 10×10, with exact fraction answers.
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