# حاسبة المصفوفات

> المحدّد والمعكوس والرتبة والصورة السلمية الصفية المختزلة لمصفوفة حتى ⁦6 × 6⁩، مع عرض كل عملية صفية، إضافة إلى المنقول وحاصل الضرب والمجموع، بكسور دقيقة.

النسخة التفاعلية: https://www.calcopenly.com/ar/math/matrix-calculator
الموضوع: حاسبات الرياضيات

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.

## المدخلات

- **Calculate** (الخيارات: 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**

## النتائج

- النتيجة — النتيجة الرئيسية
- det A
- rank A
- trace A

## الصيغة

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

## أمثلة محلولة

### Inverse of the default 3 × 3

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

## الأسئلة

### 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.

### ما مدى دقة «⁨حاسبة المصفوفات⁩»؟

تعتمد الدقة على مدخلاتك وافتراضات الطريقة. يستخدم الحساب العشري 50 رقمًا معنويًا، لكن التقديرات والأساليب العددية وبيانات المصدر قد تكون أقل دقة؛ تقريب القيم المعروضة لا يزيل هذه الحدود. أمثلة محلولة جرى التحقق منها بمصادر مستقلة: 8. مثلًا، يجري التحقق من «⁨Inverse of the default 3 × 3⁩» بالرجوع إلى ⁨Python fractions Gauss–Jordan on [A | I]; det by cofactor expansion = −1⁩.

### ما مصدر هذه الطريقة؟

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

## المصادر

- [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