# 矩阵计算器

> 计算最大 6 × 6 矩阵的行列式、逆矩阵、秩和行最简形，展示每一步行运算；同时支持转置、乘积与求和，结果使用精确分数。

交互版本：https://www.calcopenly.com/zh/math/matrix-calculator
主题：数学计算器

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.

## 输入

- **Calculate** (选项：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**

## 结果

- 结果 — 主要结果
- det A
- rank A
- trace A

## 公式

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

## 计算示例

### Inverse of the default 3 × 3

- Calculate: Inverse of A
- Matrix A: 2 1 1 / 1 3 2 / 1 0 0
- **结果: [0, 0, 1; −2, 1, 3; 3, −1, −5]**
- **det A: -1**
- 核验来源：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
- **结果: [3/5, −7/10; −1/5, 2/5]**
- **det A: 10**
- 核验来源：(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
- **结果: −1**
- **det A: -1**
- **trace A: 5**
- 核验来源：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
- **结果: 2**
- **rank A: 2**
- **det A: 0**
- 核验来源：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
- **结果: [19, 22; 43, 50]**
- 核验来源：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
- **结果: [1, 0, 0, −8; 0, 1, 0, 1; 0, 0, 1, −2]**
- **rank A: 3**
- 核验来源：Wikipedia “Gaussian elimination” (row reduction example); Python fractions RREF

## 常见问题

### 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.

### “矩阵计算器”有多准确？

准确性取决于输入值和方法的假设。十进制运算使用50位有效数字，但估算、数值方法和源数据的精度可能较低；显示时的舍入并不能消除这些限制。 已按独立来源核验的计算示例：8。 例如，“Inverse of the default 3 × 3”根据Python fractions Gauss–Jordan on [A | I]; det by cofactor expansion = −1进行核验。

### 这种方法出自哪里？

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

## 来源

- [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