Matrix Multiplication Calculator

Compute the matrix product \(C = AB\) for real or complex matrices.
Supports fractions, decimals, and mathematical expressions.

For matrices \(A\) (size \(m \times p\)) and \(B\) (size \(p \times n\)), the product \(C = AB\) is defined as:

\[ C_{ij} = \sum_{k=1}^{p} A_{ik}B_{kj} \]

Key Properties:

  • Compatibility: Number of columns in \(A\) must equal number of rows in \(B\)
  • Result size: If \(A\) is \(m \times p\) and \(B\) is \(p \times n\), then \(AB\) is \(m \times n\)
  • Non-commutative: Generally \(AB \neq BA\)
  • Associative: \((AB)C = A(BC)\)
  • Distributive: \(A(B + C) = AB + AC\)

Applications:

  • Composing linear transformations
  • Solving systems of equations
  • Graph theory (adjacency matrices)
  • Quantum mechanics (operator composition)
Enter Matrix Dimensions

Integers 1–20. Cols of A must equal Rows of B. Press Enter or click Generate ↵.

1 — Set Up Matrices
Loads a pre-filled 2×3 matrix A and 3×2 matrix B — ready to compute.
2 — Fill with Random Values
Fills both matrices with random integers from −10 to 10.
Fills both matrices with random complex entries, e.g. 3+2i.
3 — Compute
Multiplies A and B and shows result C with step-by-step dot products.
Resets all dimensions, matrix entries, and results.
Display values as: