# Matrix calculator

> Determinant, inverse, rank and RREF of a matrix up to 6 × 6 with every row operation shown, plus transpose, product and sum, in exact fractions.

Interactive version: https://www.calcopenly.com/math/matrix-calculator
Subject: Math calculators

The calculator works on matrices up to 6 × 6 in exact fractions. The inverse and the reduced row echelon form come from Gauss–Jordan elimination: [A | I] is row-reduced until the left half is the identity, and the right half is then A⁻¹. The determinant is the product of the pivots from forward elimination, negated once for each row swap; the rank is the number of pivots; and a product has entries (AB)ᵢⱼ = Σₖ aᵢₖbₖⱼ.

Linear algebra courses, 3D graphics transforms and systems of equations are the usual uses. The default matrix [2 1 1; 1 3 2; 1 0 0] has determinant −1, so it is invertible, and its inverse [0 0 1; −2 1 3; 3 −1 −5] has whole-number entries.

Type one row per line, with entries separated by spaces or commas; fractions such as 1/2 stay exact. A matrix with determinant 0 is singular: it has no inverse, and its rank is below its size.

## Inputs

- **Calculate** (options: Determinant of A, Inverse of A, Rank of A, Reduced row echelon form of A, Transpose of A, A × B, A + B, A − B)
- **Matrix A**: One row per line (or rows separated by ;), entries separated by spaces or commas; fractions like 1/2 work
- **Matrix B**

## Results

- Result — main result
- det A
- rank A
- trace A

## Formula

$$
\begin{gathered} A^{-1}:\ [A \mid I] \xrightarrow{\text{row operations}} [I \mid A^{-1}] \\[10pt] (AB)_{ij} = \sum_k a_{ik} b_{kj} \end{gathered}
$$

## Worked examples

### Inverse of the default 3 × 3

- Calculate: Inverse of A
- Matrix A: 2 1 1 / 1 3 2 / 1 0 0
- **Result: [0, 0, 1; −2, 1, 3; 3, −1, −5]**
- **det A: -1**
- Checked against: Python fractions Gauss–Jordan on [A | I]; det by cofactor expansion = −1

### Inverse of [4 7; 2 6]

- Calculate: Inverse of A
- Matrix A: 4 7 / 2 6
- **Result: [3/5, −7/10; −1/5, 2/5]**
- **det A: 10**
- Checked against: (1/(ad − bc))·[d −b; −c a] = (1/10)·[6 −7; −2 4]

### Determinant of the default 3 × 3

- Calculate: Determinant of A
- Matrix A: 2 1 1 / 1 3 2 / 1 0 0
- **Result: −1**
- **det A: -1**
- **trace A: 5**
- Checked against: Cofactor expansion: 2·0 − 1·(0 − 2) + 1·(0 − 3) = −1

### Rank of a singular matrix (edge case)

- Calculate: Rank of A
- Matrix A: 1 2 3 / 2 4 6 / 1 1 1
- **Result: 2**
- **rank A: 2**
- **det A: 0**
- Checked against: Row 2 = 2 × row 1; Python fractions RREF has 2 pivots

### [1 2; 3 4] × [5 6; 7 8]

- Calculate: A × B
- Matrix A: 1 2 / 3 4
- Matrix B: 5 6 / 7 8
- **Result: [19, 22; 43, 50]**
- Checked against: Row-by-column products by hand (e.g. 1·5 + 2·7 = 19)

### RREF of the Wikipedia augmented matrix

- Calculate: Reduced row echelon form of A
- Matrix A: 1 2 -1 -4 / 2 3 -1 -11 / -2 0 -3 22
- **Result: [1, 0, 0, −8; 0, 1, 0, 1; 0, 0, 1, −2]**
- **rank A: 3**
- Checked against: Wikipedia “Gaussian elimination” (row reduction example); Python fractions RREF

## Questions

### How do you find the inverse of a matrix?

Write A next to the identity matrix, [A | I], and apply row operations until the left half becomes I; the right half is then A⁻¹. A 2 × 2 matrix [a b; c d] has a shortcut: A⁻¹ = (1/(ad − bc)) × [d −b; −c a]. For [4 7; 2 6], ad − bc = 24 − 14 = 10, so A⁻¹ = [3/5 −7/10; −1/5 2/5].

### How do you calculate the determinant of a 3 × 3 matrix?

Expand along a row or column, multiplying each entry by the determinant of its 2 × 2 minor with alternating signs. For [2 1 1; 1 3 2; 1 0 0], the bottom row is quickest because two of its entries are 0: det = 1 × (1 × 2 − 1 × 3) = −1. For larger matrices row reduction reaches the same answer with far fewer operations.

### When does a matrix have no inverse?

When its determinant is 0, which happens exactly when one row or column is a combination of the others. In [1 2 3; 2 4 6; 1 1 1], row 2 is twice row 1, so the rank is 2 rather than 3 and the determinant is 0. Such a matrix is called singular, and a system Ax = b built on it has either no solution or infinitely many.

### How do you multiply two matrices?

Each entry of AB is a row of A times a column of B, summed: (AB)ᵢⱼ = Σₖ aᵢₖbₖⱼ. For [1 2; 3 4] × [5 6; 7 8] the top-left entry is 1 × 5 + 2 × 7 = 19, and the product is [19 22; 43 50]. A needs as many columns as B has rows, and order matters: here BA = [23 34; 31 46].

### What is reduced row echelon form?

A matrix is in reduced row echelon form (RREF) when each non-zero row starts with a 1, that leading 1 is the only non-zero entry in its column, the leading 1s step right going down, and zero rows sit at the bottom. Every matrix has exactly one RREF, and its number of leading 1s is the rank. For an augmented matrix it reads off the solution: [1 0 0 −8; 0 1 0 1; 0 0 1 −2] means x = −8, y = 1, z = −2.

### How accurate is the matrix calculator?

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 8 worked examples whose answers come from independent sources; for example, “Inverse of the default 3 × 3” is checked against Python fractions Gauss–Jordan on [A | I]; det by cofactor expansion = −1.

### Where does the method come from?

Wikipedia — Gaussian elimination (row reduction and the RREF example); Wolfram MathWorld — Matrix Inverse; G. Strang, Introduction to Linear Algebra (5th ed.), chapters 2–3.

## Sources

- [Wikipedia — Gaussian elimination (row reduction and the RREF example)](https://en.wikipedia.org/wiki/Gaussian_elimination)
- [Wolfram MathWorld — Matrix Inverse](https://mathworld.wolfram.com/MatrixInverse.html)
- G. Strang, Introduction to Linear Algebra (5th ed.), chapters 2–3