Matrix Multiplication
Multiply 2×2 matrices with our efficient calculator
Matrix multiplication is a binary operation that takes two matrices and produces another matrix.
Key Rules:
- Matrix A columns must equal Matrix B rows
- Not commutative: A × B ≠ B × A
- Result has dimensions: (A rows) × (B columns)
Formula: For 2×2 matrices, each element is calculated as the sum of products of corresponding row and column elements.
When can two matrices be multiplied?
Two matrices can be multiplied when the number of columns in the first matrix equals the number of rows in the second matrix. If A is m x n and B is n x p, the result AB will be m x p. For example, a 2x3 matrix can multiply a 3x4 matrix, giving a 2x4 result.
Is matrix multiplication commutative?
No, matrix multiplication is NOT commutative. In general, AB does not equal BA. In fact, if AB exists, BA might not even be defined (different dimensions). Even when both products exist, they usually give different results. This non-commutativity is crucial in many applications.
What is the identity matrix?
The identity matrix I is a square matrix with 1s on the diagonal and 0s elsewhere. It acts as the multiplicative identity: AI = IA = A for any compatible matrix A. It's analogous to multiplying by 1 in regular arithmetic.
Where is matrix multiplication used?
Matrix multiplication is fundamental in computer graphics (transformations, rotations), machine learning (neural networks), physics (quantum mechanics), economics (input-output analysis), solving systems of linear equations, and representing any linear transformation between vector spaces.