Matrix Multiplication Calculator

Matrix multiplication combines two matrices to produce a new matrix. The element in row i and column j of the result is the dot product of row i from the first matrix and column j from the second matrix. Unlike number multiplication, matrix multiplication is not commutative: A × B generally does not equal B × A.

To multiply matrix A (dimensions m × n) by matrix B (dimensions n × p), the number of columns in A must equal the number of rows in B. The result is an m × p matrix. For example, a 2x3 matrix can be multiplied by a 3x4 matrix, but not by a 4x3 matrix.

No. Matrix multiplication is generally not commutative, meaning A × B ≠ B × A in most cases. This is one of the key differences between matrix multiplication and scalar multiplication. The order matters fundamentally in matrix operations.

The element in row i, column j of A × B is calculated by taking row i from A and column j from B, multiplying corresponding elements, and summing the results. This is the dot product of a row vector and a column vector.

Matrix multiplication is essential in computer graphics (combining transformations), physics (representing operations), machine learning (neural networks), and solving systems of linear equations. Essentially any multi-dimensional linear transformation uses matrix multiplication.