# Réponse finale obtenue du bot:
response = « Here’s a comprehensive exploration of the properties of algebra for equations and inequalities, polynomials and functions, linear algebra and matrices:
**Équations et Inéquations**
1. **Symmetry**: An equation remains unchanged when its variables are swapped.
2. **Addition and Subtraction Property**: Adding or subtracting the same value to both sides of an equation leaves the equation true.
3. **Multiplication and Division Property**: Multiplying or dividing both sides of an equation by a non-zero value leaves the equation true.
4. **Transitive Property**: If `a = b` and `b = c`, then `a = c`.
5. **Reflexive Property**: An equation remains true when its variables are equal to themselves.
**Polynômes et Fonctions**
1. **Addition Property of Polynomials**: `(a + b) = a + b`
2. **Multiplication Property of Polynomials**: `(a * b) = ab`
3. **Distributive Property**: `a(b + c) = ab + ac`
4. **Substitution Property**: Replacing a variable with an expression in an equation leaves the equation true.
5. **Composition Property**: `(f ∘ g)(x) = f(g(x))`
**Algèbre Linéaire et Matrices**
1. **Associative Property of Addition**: `(a + b) + c = a + (b + c)`
2. **Commutative Property of Addition**: `a + b = b + a`
3. **Distributive Property of Matrix Multiplication**: `A(B + C) = AB + AC`
4. **Invertible Property**: A matrix has an inverse if and only if its determinant is non-zero.
5. **Linear Independence**: A set of vectors is linearly independent if none can be expressed as a linear combination of the others.
These properties are fundamental to algebra and play a crucial role in solving equations, inequalities, and systems of linear equations. Understanding these properties helps students develop problem-solving skills and appreciate the beauty of algebra. »