What Are the Rules for Matrix Multiplication?
A(m×n) × B(n×p) = C(m×p). The inner dimensions must match: columns of A must equal rows of B. Element C[i][j] = sum of A[i][k] × B[k][j] for all k from 1 to n.
The Dimension Rule
For multiplication to be defined, the number of columns in A must equal the number of rows in B:
A: m x [n] * B: [n] x p = C: m x p
^ ^
must matchThe result matrix C has the outer dimensions: rows of A and columns of B.
The Element Formula
Each element of the result is a dot product of a row from A and a column from B:
C[i][j] = A[i][1]*B[1][j] + A[i][2]*B[2][j] + ... + A[i][n]*B[n][j]Worked Example
A = | 1 2 3 | B = | 7 8 |
| 4 5 6 | | 9 10 |
| 11 12 |
A is 2x3, B is 3x2, so C = A*B is 2x2.Computing each element:
- C[1][1] = 1×7 + 2×9 + 3×11 = 7 + 18 + 33 = 58
- C[1][2] = 1×8 + 2×10 + 3×12 = 8 + 20 + 36 = 64
- C[2][1] = 4×7 + 5×9 + 6×11 = 28 + 45 + 66 = 139
- C[2][2] = 4×8 + 5×10 + 6×12 = 32 + 50 + 72 = 154
C = | 58 64 |
| 139 154 |Key Properties
| Property | Holds? |
|---|---|
| Associative: (AB)C = A(BC) | Yes |
| Distributive: A(B+C) = AB + AC | Yes |
| Commutative: AB = BA | No (in general) |
| Identity: AI = IA = A | Yes |
Use the ML3X matrix calculator to multiply matrices up to 5×5 with full dot-product step-by-step breakdown for every element.