---
title: "Matrix Multiplication Calculator — General Dimensions (2×2 to 4×4)"
description: "Multiply two matrices of general dimensions (up to 4×4) with step-by-step results. Shows C[i][j] dot product calculation. For linear algebra, machine learning, and engineering."
url: "https://worldcalculators.org/calculators/matrix-multiply/"
canonical: "https://worldcalculators.org/calculators/matrix-multiply/"
---

# Matrix Multiplication Calculator

Multiply two matrices of general dimensions (up to 4×4) with step-by-step results. Shows C[i][j] dot product calculation. For linear algebra, machine learning, and engineering.

## Quick answer

Matrix Dimensions

## Why Matrix Multiplication Matters

Every neural network forward pass is essentially a sequence of matrix multiplications. Google's search ranking uses matrix operations. Computer graphics transformations (rotate, scale, translate) are matrix multiplications chained together. Understanding matrix multiplication is central to modern computing.

## Frequently Asked Questions

Naive matrix multiplication of two n×n matrices takes O(n³) operations. Strassen's algorithm (1969) reduces this to O(n^2.807). The current best theoretical algorithm is O(n^2.371). For practical use, libraries like NumPy, BLAS, and cuBLAS use highly optimized algorithms with cache-friendly memory access patterns and SIMD instructions.
Every matrix multiplication represents the composition of two linear transformations. If T₁ is represented by A and T₂ by B, then applying T₁ then T₂ corresponds to computing B×A (not A×B — note the order reversal). This is why order matters: "first rotate, then scale" gives a different result than "first scale, then rotate."

Source: https://worldcalculators.org/calculators/matrix-multiply/
Markdown mirror of the HTML page. Prefer this URL for RAG; the interactive calculator still lives on the HTML page.