AlgebraFree

Matrix Multiplication Calculator

Multiply two 2x2 matrices from numeric grid input.

Enter Values

A11

A12

A21

A22

B11

B12

B21

B22

Result

Enter values above and click Calculate to see your result.

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Formula

Core Formula

(AB)_{ij} = \sum_k A_{ik} B_{kj}

How it works: Each output entry is a row by column dot product.

Worked Example

Enter A and B values and compute AB.

How Matrix Multiplication Works

Matrix multiplication combines two matrices by taking dot products of rows from the first matrix with columns of the second. Each entry in the result is one such dot product.

Entry (i,j) of the product AB equals the dot product of row i of A with column j of B
Matrix multiplication is NOT commutative: AB is generally different from BA
For two 2x2 matrices, the result is always another 2x2 matrix
Each entry requires 2 multiplications and 1 addition for 2x2, giving 8 multiplications and 4 additions total

Matrix multiplication is the foundation of linear transformations, computer graphics, machine learning, and many areas of applied mathematics.

You can also calculate changes using our Matrix Determinant Calculator, Inverse Matrix Calculator, Matrix Transpose Calculator or 2x2 Determinant Calculator.

Frequently Asked Questions

Does the order of multiplication matter?

Yes. Matrix multiplication is not commutative, meaning AB usually does not equal BA. Always specify which matrix is on the left and which is on the right.

What size is the output?

For two 2x2 matrices, the product is always 2x2. In general, multiplying an m x n matrix by an n x p matrix gives an m x p result.

How is each entry calculated?

Each entry (i,j) in the product is the dot product of row i from the first matrix with column j from the second matrix. For 2x2, that means multiplying two pairs and adding.

Can I multiply matrices of different sizes?

This calculator handles 2x2 only. In general, the number of columns in the first matrix must equal the number of rows in the second matrix for multiplication to be defined.

Is it possible to embed the Matrix Multiplication Calculator on another website?

Yes, embedding the Matrix Multiplication Calculator is free. Hit the "Embed" button on this page, adjust the width, height, and theme, then grab the iframe code. It works on WordPress, Wix, Squarespace, Shopify, and plain HTML pages. No registration needed. Full instructions at calculory.com/services/embed-calculators.

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Ask about this calculator

I can help you understand the matrix multiplication calculator formula, interpret your results, and answer follow-up questions.

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