A fundamental operation in linear algebra that combines two matrices to produce a third matrix by multiplying rows of the first with columns of the second.

Matrix Multiplication

Matrix multiplication is a crucial linear algebra operation that defines how to combine two matrices to create a resulting matrix. Unlike simple arithmetic multiplication, matrix multiplication follows specific rules and properties that make it both powerful and distinct.

Definition and Rules

To multiply two matrices:

Matrix A must have the same number of columns as Matrix B has rows
The resulting matrix will have dimensions (rows of A × columns of B)
Each element is calculated through dot product operations

For matrices A(m×n) and B(n×p), the resulting matrix C(m×p) is calculated as:

C[i,j] = Σ(A[i,k] * B[k,j]) for k = 1 to n

Properties

Non-commutativity: Unlike regular multiplication, AB ≠ BA (in general)
Associativity: (AB)C = A(BC)
Distributivity: A(B+C) = AB + AC
Identity Matrix: AI = IA = A, where I is the identity matrix

Applications

Matrix multiplication appears in numerous fields:

Linear transformations in computer graphics
Neural networks weight calculations
Systems of equations solving
Computer vision operations
Quantum mechanics calculations

Computational Considerations

The standard algorithm for matrix multiplication has a time complexity of O(n³), making it computationally intensive for large matrices. Several optimized algorithms exist:

Special Cases

Square Matrices: When both matrices are n×n
Vector Multiplication: When one matrix is 1×n (vector)
Diagonal Matrices: Simplified multiplication rules apply
Sparse Matrices: Special algorithms for matrices with many zeros

Matrix multiplication serves as a cornerstone for many advanced mathematical concepts and has widespread applications in scientific computing, making it essential for modern computational tasks and mathematical modeling.