Matrix Multiplication — Rules, Steps & Examples

Matrix Multiplication Step by Step

Matrix multiplication is fundamentally different from addition. To compute C = A * B, each element C[i,j] is the dot product of the i-th row of A and the j-th column of B:

C[i,j] = sum(A[i,k] * B[k,j]) for k = 1 to n

Dimension Requirements

For A (m x n) and B (p x q):

• n must equal p (A's columns must equal B's rows)
• The result C has dimensions m x q

Example: 2x2 Multiplication

A = | 1  2 |    B = | 5  6 |
    | 3  4 |        | 7  8 |

C[1,1] = 1*5 + 2*7 = 5 + 14 = 19
C[1,2] = 1*6 + 2*8 = 6 + 16 = 22
C[2,1] = 3*5 + 4*7 = 15 + 28 = 43
C[2,2] = 3*6 + 4*8 = 18 + 32 = 50

C = | 19  22 |
    | 43  50 |

Key Properties

• Not commutative: A * B does not generally equal B * A
• Associative: (A * B) * C = A * (B * C)
• Distributive: A * (B + C) = A * B + A * C
• Identity: A * I = I * A = A

Why Order Matters

Consider a 2x3 matrix A and a 3x4 matrix B. A * B gives a 2x4 matrix, but B * A is undefined since B has 4 columns and A has 2 rows. Even for square matrices, changing the order usually produces different results.