Matrix Multiplication

Matrix Setup

Matrix A 2 × 2

Rows × Cols

Matrix B 2 × 2

Rows

× Cols

Examples

Both matrices must be square and the same size. B rows/cols are locked to A.

Size n×n

Matrix A 2 × 2

Matrix B 2 × 2

Examples

Result

Enter matrix values above and press Multiply to see the result and step-by-step dot product computations.

Result C = A × B

A × B

B × A

Dimension Rule

A (m×n) × B (n×p) = C (m×p)

For A×B to be defined, A's column count must equal B's row count — the inner dimensions must match. The result C has the outer dimensions.

Think of it this way: the shared dimension n is "consumed" during multiplication. What remains are the outer dimensions m (rows of A) and p (cols of B).

To find entry C_ij: take the dot product of row i of A with column j of B — multiply matching entries and add them all up.

C_ij = a_i1·b_1j + a_i2·b_2j + … + a_in·b_nj

AB ≠ BA in General

AB ≠ BA (not commutative)

Matrix multiplication is not commutative — swapping the order almost always changes the result, and sometimes AB is defined while BA is not (when dimensions don't match in reverse).

However, matrix multiplication is associative and distributive:

(AB)C = A(BC) — associative: grouping doesn't matter.

A(B+C) = AB + AC — distributive over addition.

AB is defined only when cols(A) = rows(B)
In general, AB ≠ BA even when both are square
(AB)ᵀ = BᵀAᵀ — transpose reverses order
Identity matrix I satisfies AI = IA = A

Matrix multiplication still confusing?