Augmented matrix solver
Type the augmented matrix [A|b] and learn whether the system has one solution, infinitely many or none; the classic three-equation example returns the solution (2, 3, −1). You also see the rank and the reduced form. Free in your browser.
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An augmented matrix packs a linear system Ax equals b into a single array [A|b] so row operations act on coefficients and constants together. Students writing Gauss-Jordan by hand, tutors checking homework, and automation scripts that already store systems as augmented matrices all need a deterministic answer: a unique solution vector when the rank of A matches the number of variables, a clear infinite-solutions signal when free variables remain, or a no-solution flag when a reduced row is all zeros on the left and nonzero on the right. This augmented matrix solver accepts that matrix as a rectangular array of finite numbers, reduces it with partial pivoting, and returns status, rank, RREF, and the solution when it is unique.
What an augmented matrix encodes and how reduction decides the case
A linear system with m equations in n unknowns is written Ax equals b, where A is m by n and b is a length-m column. Stacking b as an extra column produces the augmented matrix [A|b] with m rows and n plus one columns. Elementary row operations—swap two rows, scale a row by a nonzero constant, or add a multiple of one row to another—do not change the solution set of the system.
Gauss-Jordan elimination with partial pivoting systematically creates leading ones (pivots) and clears every other entry in each pivot column until the matrix is in reduced row echelon form. After reduction, three mutually exclusive outcomes appear. If any row is all zeros on the coefficient side and a nonzero constant on the right, the system is inconsistent and has no solution.
If every nonzero row has a pivot in a coefficient column and the number of pivots equals n, each variable is bound and there is a unique solution read from the last column. If the system is consistent but the number of pivots is less than n, at least one free variable remains and there are infinitely many solutions.
Where an augmented matrix solver earns its keep
Linear algebra courses introduce systems through augmented matrices long before matrix-inverse or Cramer workflows, so learners search for an augmented matrix solver that mirrors the [A|b] notation in their notes. Engineering notebooks that already assemble constraint blocks as augmented rows avoid splitting A and b just to call a square-only solver; rectangular and over- or under-determined systems are first-class here. Failed validation does not consume balance.
Keep requests within the row and column limits, pin the rounded RREF and solution in tests, and you get reproducible classifications for homework, exam generators, and edge workers without standing up a full computer-algebra stack of your own.
What you can do with it
Classroom Gauss-Jordan checks
Students paste their [A|b] matrix and compare status, rank, RREF, and unique solution against hand reduction—including infinite and inconsistent cases.
LMS and exam auto-grading
Score linear-system items by matching the ordered solution vector when unique, or the infinite/none classification when the system is singular or inconsistent.
Notebooks with rectangular systems
Solve or classify over- and under-determined blocks stored as augmented rows without requiring a square coefficient matrix or a separate b vector.
FAQ
How do I solve a system from an augmented matrix?
Send matrix as rows of [A|b] (finite numbers, equal row length).
When is status none returned?
When RREF contains a row that is all zeros in the coefficient columns and a nonzero right-hand side—the classic inconsistent [0 … 0 | c] pattern. There is no solution.
Must the matrix be square?
No. Rows and columns may differ (within limits). Square systems with a full set of pivots yield a unique solution; rectangular systems may be unique, infinite, or inconsistent depending on rank.
What does it mean when the system has infinitely many solutions?
The system is consistent but has fewer pivots than variables, so at least one variable can take any value. The tool tells you so and shows the reduced matrix, where you can see which variables are left open.
How large a matrix can I enter?
Up to 32 rows and 33 columns, which covers 32 equations in 32 unknowns. Every row needs at least two numbers, one coefficient and the right-hand side, and all rows must have the same length.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/algebra/augmented-matrix-solver \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"matrix":[[2,1,-1,8],[-3,-1,2,-11],[-2,1,2,-3]]}'const res = await fetch("https://api.kit.forhosting.com/algebra/augmented-matrix-solver", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"matrix": [
[
2,
1,
-1,
8
],
[
-3,
-1,
2,
-11
],
[
-2,
1,
2,
-3
]
]
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/algebra/augmented-matrix-solver",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"matrix": [
[
2,
1,
-1,
8
],
[
-3,
-1,
2,
-11
],
[
-2,
1,
2,
-3
]
]
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/algebra/augmented-matrix-solver", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"matrix":[[2,1,-1,8],[-3,-1,2,-11],[-2,1,2,-3]]}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"matrix":[[2,1,-1,8],[-3,-1,2,-11],[-2,1,2,-3]]}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/algebra/augmented-matrix-solver", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"matrix": [
[
2,
1,
-1,
8
],
[
-3,
-1,
2,
-11
],
[
-2,
1,
2,
-3
]
]
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "algebra.augmented_matrix_solver",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
What an augmented matrix encodes and how reduction decides the case
This endpoint reports those three cases under the status field unique, infinite, or none, and attaches the cleaned RREF so you can show the same intermediate form a textbook would print after hand reduction.
Input contract, response shape, and validation rules
Every row must have at least two entries (one coefficient column plus the right-hand side). At most thirty-two rows and thirty-three columns are accepted so edge workers stay bounded. When status is none, solution is omitted and the message explains the inconsistent zero-left nonzero-right row. Example: matrix equals [[2,1,-1,8],[-3,-1,2,-11],[-2,1,2,-3]] is the classic three-variable system with unique solution [2,3,-1]. Pair the response with your own graphing or step log if you need animated row ops; this capability returns the classification and RREF, not a prose walkthrough of every swap. POST a JSON body with the required field matrix: an array of one or more rows, each row an array of finite numbers of equal length. Aliases augmented, a, A, and mat are accepted for the same field; the matrix may also arrive as a JSON string of that array. Ragged rows, empty matrices, non-finite entries, and oversized shapes fail with invalid_input naming the problem, and those failures are not billed. On success the body always includes status (unique, infinite, or none), num_equations, num_variables, rank (number of pivots in the coefficient block), rref (the reduced augmented matrix with stable rounding), and a short human message. When status is unique, solution is an array of length num_variables giving x in variable order x1 through xn. When status is infinite, free_variables counts how many variables lack a pivot; the particular solution vector is omitted because many parametrizations are equally valid.
Where an augmented matrix solver earns its keep
Tutoring platforms and LMS plugins can POST student matrices and grade unique solutions against the ordered solution array while treating infinite and none as distinct correct classifications rather than silent empty lists.
Technical questions
What does status infinite mean?
The reduced system is consistent but rank is less than the number of variables, so free variables remain and there are infinitely many solutions. free_variables counts them; no single solution vector is returned.
What does the augmented matrix solver cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.