# Linear Algebra Rank, Nullity & RREF

> Linear Algebra Rank, Nullity & RREF is a paid API for AI agents from linear-algebra.x402tookit.com, paid per call via x402, $0.02/call, status unknown (last checked 2026-10-29).

Computes the rank, nullity, reduced row echelon form (RREF), and pivot columns of a matrix up to 10x10

## Facts

- Endpoint: GET https://linear-algebra.x402tookit.com/rank?utm_source=zero.xyz
- Price: $0.02/call
- Payment: x402
- Status: unknown
- Last checked: 2026-10-29
- Activations on Zero: 0
- Tags: x402
- Canonical page: https://www.zero.xyz/c/linear-algebra-rank-nullity-rref-6928b66e
- Structured record (JSON): https://api.zero.xyz/v1/capabilities/cap_XWileAqS7qe2jbiBjlTlF

Status and success rate cover calls made through Zero and Zero's own probes. Third-party monitors may report differently.

## How to call it through Zero

Zero handles the 402 payment challenge and records the run. With the Zero CLI installed (`npm i -g @zeroxyz/cli`):

```sh
zero fetch --capability linear-algebra-rank-nullity-rref-6928b66e
```

Example prompt: What's the rank, nullity, and reduced row echelon form of the matrix [[1,2,3],[4,5,6],[7,8,9]]?

## When to prefer this

Use this endpoint when you need the rank, nullity, pivot columns, or RREF of a matrix in a single call. It is preferable over generic CAS tools when you need structured JSON output including all four properties at once, and is especially useful for educational, debugging, or pre-processing steps before solving linear systems.

## Known failure modes

- Matrix exceeds 10x10 size limit — returns error
- Malformed matrix input (invalid JSON or semicolon format) — returns parse error
- Non-numeric values in matrix — returns validation error
- Service unavailable or payment failure — HTTP error with no result

## How this service works

Rank, nullity and reduced row echelon form of a matrix (up to 10x10) with pivot columns.

## Output

Returns the matrix rank, nullity, left nullity, reduced row echelon form (RREF) as a 2D array, pivot column indices, a full-rank boolean flag, row and column dimensions, a summary string, and optional degraded/disclaimer fields.

## Request schema (JSON Schema)

```json
{
 "type": "object",
 "$schema": "https://json-schema.org/draft/2020-12/schema",
 "required": [
  "input"
 ],
 "properties": {
  "input": {
   "type": "object",
   "required": [
    "type",
    "method"
   ],
   "properties": {
    "type": {
     "type": "string",
     "const": "http"
    },
    "method": {
     "enum": [
      "GET"
     ],
     "type": "string"
    },
    "queryParams": {
     "type": "object",
     "required": [
      "A"
     ],
     "properties": {
      "A": {
       "type": "string",
       "description": "Matrix, JSON like [[1,2,3],[4,5,6],[7,8,9]] or 1,2,3;4,5,6;7,8,9 (max 10x10)"
      }
     }
    }
   },
   "additionalProperties": false
  },
  "output": {
   "type": "object",
   "required": [
    "type"
   ],
   "properties": {
    "type": {
     "type": "string"
    },
    "example": {
     "type": "object",
     "properties": {
      "asOf": {
       "type": [
        "null",
        "string"
       ]
      },
      "cols": {
       "type": [
        "null",
        "number"
       ]
      },
      "rank": {
       "type": [
        "null",
        "number"
       ]
      },
      "rows": {
       "type": [
        "null",
        "number"
       ]
      },
      "rref": {
       "type": [
        "array",
        "null"
       ],
       "items": {
        "type": "array",
        "items": {
         "type": "number"
        }
       }
      },
      "source": {
       "type": [
        "null",
        "string"
       ]
      },
      "nullity": {
       "type": [
        "null",
        "number"
       ]
      },
      "reasons": {
       "type": [
        "array",
        "null"
       ],
       "items": {
        "type": "string"
       }
      },
      "summary": {
       "type": [
        "null",
        "string"
       ]
      },
      "verdict": {
       "type": [
        "null",
        "string"
       ]
      },
      "degraded": {
       "type": [
        "boolean",
        "null"
       ]
      },
      "fullRank": {
       "type": [
        "boolean",
        "null"
       ]
      },
      "disclaimer": {
       "type": [
        "null",
        "string"
       ]
      },
      "notChecked": {
       "type": [
        "array",
        "null"
       ],
       "items": {
        "type": "string"
       }
      },
      "leftNullity": {
       "type": [
        "null",
        "number"
       ]
      },
      "pivotColumns": {
       "type": [
        "array",
        "null"
       ],
       "items": {
        "type"
… (truncated)
```

## More

- Live health (JSON, refreshed every minute): https://www.zero.xyz/c/linear-algebra-rank-nullity-rref-6928b66e/health.json
- [Zero catalog index](https://www.zero.xyz/llms.txt)
- [Other services from linear-algebra.x402tookit.com](https://www.zero.xyz/host/linear-algebra.x402tookit.com/llms.txt)