Matrix Multiplication Calculator

About Matrix Multiplication Calculator

The Matrix Multiplication Calculator allows you to multiply two matrices together and see every step of the computation. Each element of the result matrix is calculated as the dot product of a row from Matrix A and a column from Matrix B. This calculator supports matrices up to 5×5, provides interactive highlighting so you can see exactly which row and column produce each result element, and displays the full mathematical working using MathJax-rendered formulas.

How Matrix Multiplication Works

Given Matrix A of size m×n and Matrix B of size n×p, the product C = A × B is a matrix of size m×p. Each element is computed as:

$$C[i,j] = \sum_{k=1}^{n} A[i,k] \times B[k,j]$$

This means you take the i-th row of A and the j-th column of B, multiply corresponding elements, and sum all the products. This operation is called the dot product.

Key Properties of Matrix Multiplication

How to Use the Matrix Multiplication Calculator

1. Set dimensions — Choose the rows and columns for Matrix A and the columns for Matrix B. The number of columns in A automatically sets the number of rows in B.
2. Enter values — Type numbers into each cell. Use the quick examples for preset matrices.
3. Calculate — Click "Multiply A × B" to see the result matrix and step-by-step breakdown.
4. Explore results — Hover over or click any result cell to see its dot product visualized with color-coded highlighting. Use "Play All" to auto-step through every element.

Dimension Compatibility Rule

Real-World Applications

Frequently Asked Questions

What is matrix multiplication?

Matrix multiplication is an operation that takes two matrices A (m×n) and B (n×p) and produces a result matrix C (m×p). Each element C[i][j] is computed as the dot product of the i-th row of A and the j-th column of B.

Why must the number of columns in A equal the number of rows in B?

For the dot product to be defined, the two vectors being multiplied must have the same length. The row of A has n elements and the column of B has n elements, so A must have as many columns as B has rows.

Is matrix multiplication commutative?

No, matrix multiplication is not commutative. In general, A × B does not equal B × A. The result dimensions may differ, and even when both products are defined and have the same size, the values are usually different.

What is the dot product in matrix multiplication?

The dot product for element C[i][j] is computed by multiplying each element of row i of Matrix A with the corresponding element of column j of Matrix B, then summing all those products. For example, if row i is [a₁, a₂, a₃] and column j is [b₁, b₂, b₃], the dot product is a₁×b₁ + a₂×b₂ + a₃×b₃.

What is the time complexity of matrix multiplication?

The standard matrix multiplication algorithm has a time complexity of O(m × n × p) for multiplying an m×n matrix with an n×p matrix. More efficient algorithms like Strassen's algorithm can reduce this to approximately O(n²·⁸⁰⁷) for square matrices.