Matrix Multiplication
Definition of matrix product
About Matrix Multiplication
The Matrix Multiplication represents definition of matrix product. This linear algebra formula is fundamental to mathematical analysis and serves as a cornerstone concept that students and professionals encounter throughout their mathematical journey. Its importance extends beyond pure mathematics into applied fields where quantitative analysis is required.
This formula is essential in Linear algebra and Matrix theory. It serves as a building block for more advanced mathematical theory and provides the foundation needed to understand complex mathematical relationships. Whether you're studying mathematics, physics, engineering, or economics, familiarity with this formula enhances your analytical capabilities.
Practical applications of the Matrix Multiplication include Computer graphics, Machine learning, Engineering, among others. Understanding and correctly applying this formula enables problem-solvers to approach challenges more systematically and efficiently. Mastery of this concept not only expands your mathematical knowledge but also improves your overall quantitative reasoning skills.
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LaTeX Code
(AB)_{ij} = \sum_{k=1}^{n} A_{ik} B_{kj}Formula Information
Difficulty Level
Prerequisites
Discovered
19th century
Discoverer
Arthur Cayley
Real-World Applications
Examples
Mathematical Fields
Keywords
Related Topics
Important Notes
Not commutative (AB ≠ BA in general). Number of columns in A must equal rows in B.
Alternative Names
Common Usage
Formula Variations
Frequently Asked Questions
How do I multiply two matrices?
To multiply matrices A (m×n) and B (n×p), the number of columns in A must equal rows in B. The result C = AB is m×p. Element Cᵢⱼ is the dot product of row i of A and column j of B: Cᵢⱼ = Σₖ AᵢₖBₖⱼ. Multiply each element of row i with corresponding element of column j, then sum.
Why isn't matrix multiplication commutative?
Matrix multiplication is NOT commutative: AB ≠ BA in general. This is because: 1) Dimensions might not match (if AB exists, BA might not), 2) Even when both exist, the results differ. For example, if A is 2×3 and B is 3×2, then AB is 2×2 but BA is 3×3 - completely different sizes!
What are the rules for matrix multiplication?
Key rules: 1) Dimensions must match (A is m×n, B is n×p → AB is m×p), 2) Not commutative (AB ≠ BA), 3) Associative ((AB)C = A(BC)), 4) Distributive (A(B+C) = AB + AC), 5) Identity: AI = IA = A (where I is identity matrix). Always check dimensions first!
How is matrix multiplication used in applications?
Matrix multiplication is fundamental in: computer graphics (transformations, rotations), machine learning (neural networks, weight updates), engineering (system analysis, control theory), physics (quantum mechanics, transformations), economics (input-output models), and cryptography (encryption algorithms).
What's the relationship between matrix multiplication and linear transformations?
Matrix multiplication represents linear transformations. If T is a linear transformation and A is its matrix, then T(x) = Ax. Composing transformations corresponds to multiplying matrices: (T₂∘T₁)(x) = A₂(A₁x) = (A₂A₁)x. This is why matrix multiplication is defined the way it is!
How do I multiply a matrix by a vector?
Matrix-vector multiplication is a special case: if A is m×n and v is n×1, then Av is m×1. Each element of Av is the dot product of a row of A with v. This represents applying a linear transformation to the vector. For example, in 2D, a 2×2 matrix transforms a 2D vector to another 2D vector.
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Quick Details
- Category
- Linear Algebra
- Difficulty
- Intermediate
- Discovered
- 19th century
- Discoverer
- Arthur Cayley
- Formula ID
- matrix-mult