Matrix Dot Product Calculator

Matrix dot product, also called matrix multiplication, is a fundamental operation in linear algebra where two matrices are combined to produce a third matrix. For matrices A and B, the dot product A × B is defined only when the number of columns in A equals the number of rows in B. The result is a matrix with the same number of rows as A and the same number of columns as B.

Two matrices A and B can be multiplied (A × B) if and only if the number of columns in matrix A equals the number of rows in matrix B. For example, if A is a 3×4 matrix and B is a 4×2 matrix, they can be multiplied because A has 4 columns and B has 4 rows. The result will be a 3×2 matrix.

For matrices A (m×n) and B (n×p), the element at position [i,j] in the result matrix C = A × B is calculated as: C[i,j] = Σ(A[i,k] × B[k,j]) for k = 1 to n. This means you multiply each element of row i from A by the corresponding element of column j from B, then sum all the products.

No, matrix multiplication is not commutative in general. This means A × B ≠ B × A for most matrices. Even when both products are defined, they often produce different results. For example, if A is 2×3 and B is 3×2, then A × B is 2×2 but B × A is 3×3.

Matrix multiplication is used in computer graphics (transformations, rotations), machine learning (neural networks, data processing), physics (quantum mechanics, mechanics), economics (input-output models), cryptography (encryption algorithms), and many other fields where linear transformations and systems of equations are important.