Matrix calculator
Every step shown, in exact fractions.
Output
Enter a matrix: one row per line, entries separated by spaces or commas — e.g. 2 1 on the first line and 1 3 on the second. Fractions such as 3/4 are fine.
Your files stay on your device
This tool runs entirely in your browser. What you paste is processed on your device and is never sent to a server, so it is safe to use with tokens, keys and unpublished text.
Example
Example: a ticket problem solved as a 3 × 3 system, step by step
A school play sold 120 tickets for $1,180: adults pay $12, children $7, seniors $9, and twice as many children as seniors came. With x1 adults, x2 children and x3 seniors: x1 + x2 + x3 = 120, 12x1 + 7x2 + 9x3 = 1180, x2 − 2x3 = 0.
- Choose Solve a linear system and type the augmented matrix, constants last: 1 1 1 120 / 12 7 9 1180 / 0 1 -2 0.
- Step 1, R2 ← R2 − 12·R1, gives 0 −5 −3 | −260; step 2 divides that row by −5: 0 1 3/5 | 52.
- Three more steps clear the columns, with exact fractions such as −13/5 on the way, until only the identity is left.
| Unknown | Value |
|---|---|
| x1, adults | 60 |
| x2, children | 40 |
| x3, seniors | 20 |
Unique solution, in five row operations: 60 adults, 40 children and 20 seniors. The determinant of the coefficients is 13, not 0, which is why there is exactly one answer.
How it works
- Type matrix A with one row per line and the entries separated by spaces or commas — fractions like 1/3 and decimals are fine.
- Choose the operation; for A × B or A + B, type matrix B in its own box. To solve a system, put the constants in the last column.
- Read the result and every row operation that led to it, in exact fractions with decimals beside them.
Frequently asked questions
- How do I solve a system of equations?
- Write it as an augmented matrix: one row per equation, the coefficients then the constant. 2x + y − z = 8, −3x − y + 2z = −11, −2x + y + 2z = −3 is “2 1 -1 8 / -3 -1 2 -11 / -2 1 2 -3”, and gives x1 = 2, x2 = 3, x3 = −1. It also tells you when there is no solution or infinitely many, with the free variables.
- Why fractions?
- The calculation is exact: 1/3 stays 1/3 instead of 0.333…, so no rounding creeps into an inverse or a determinant. Decimals are shown beside any fraction. The inverse of “2 1 / 1 3” is [3/5 −1/5; −1/5 2/5].
- How is the determinant worked out?
- By row reduction to a triangle: each step is shown, a row swap flips the sign, and the determinant is the product of the diagonal. For “2 1 / 1 3” it is 5.
- Which formats can I paste?
- Rows on separate lines, [[1,2],[3,4]], or 1 2; 3 4 on one line. Up to 10 × 10.
- What if the matrix has no inverse?
- Then its determinant is 0 and the calculator says so: “1 2 3 / 2 4 6 / 1 1 1” is singular, because its second row is twice the first. In a system the same thing shows up as no solution or infinitely many.
- Why are the unknowns called x1, x2, x3?
- A system can have up to ten of them, so the calculator numbers the columns instead of naming them x, y, z: x1 is the first column, x2 the second. A free variable is written t1, t2 and so on.
- Is anything saved?
- No. It is calculated in your browser.
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