Matrix Multiplication Calculator

Rules of Matrix Multiplication

Unlike regular multiplication, matrix multiplication is not commutative (A × B is generally not equal to B × A), and it is subject to strict dimensional constraints. To multiply matrix A by matrix B, the matrices must be **compatible**:

• If A has dimensions m × n and B has dimensions n × p, they can be multiplied. The inner dimensions (n) must match.
• The resulting product matrix C will have dimensions m × p (the outer dimensions).

How to Calculate Matrix Multiplication

To find the entry in row i and column j of the product matrix C, you take the dot product of the i-th row of A and the j-th column of B:

C_i,j = a_i,1b_1,j + a_i,2b_2,j + ... + a_i,nb_n,j

Worked Example: 2x2 Matrix Multiplication

Let's multiply A and B:

The entries of product matrix C are computed as follows:

• Entry C_1,1 (Row 1 × Col 1): (1 × 5) + (2 × 7) = 5 + 14 = 19
• Entry C_1,2 (Row 1 × Col 2): (1 × 6) + (2 × 8) = 6 + 16 = 22
• Entry C_2,1 (Row 2 × Col 1): (3 × 5) + (4 × 7) = 15 + 28 = 43
• Entry C_2,2 (Row 2 × Col 2): (3 × 6) + (4 × 8) = 18 + 32 = 50

The resulting matrix C is:

Properties of Matrix Multiplication

• Associative: A(BC) = (AB)C
• Distributive: A(B + C) = AB + AC
• Identity Element: AI = IA = A (where I is the Identity matrix)
• Transpose of Product: (AB)^T = B^TA^T