product of matrix calculator

Enter matrix dimensions, generate input fields, and compute the matrix product A × B. Blank cells are treated as 0.

Rows in A

Columns in A

Rows in B

Columns in B

Matrix A

Matrix B

How the Product of Matrices Works

Matrix multiplication is not performed element-by-element. Instead, each value in the result matrix is created by taking a row from Matrix A and a column from Matrix B, multiplying corresponding entries, and adding those products together.

If Matrix A is m × n and Matrix B is n × p, then the product A × B exists and the result has size m × p.

Quick rule: columns of A must equal rows of B.

Step-by-Step Formula

Entry-by-entry computation

The value in row i and column j of the product matrix C is:

C_ij = Σ (A_ik × B_kj) for all valid k.

Pick row i from A
Pick column j from B
Multiply matching terms
Add them to get one cell of C

Why This Calculator Is Useful

A product of matrix calculator helps you avoid arithmetic errors and saves time when matrices become large. It is especially handy for students, engineers, data analysts, and anyone working with linear algebra.

Common use cases

Solving systems of linear equations
Computer graphics transformations
Machine learning feature transformations
Markov chains and probability modeling
Physics and engineering simulations

Tips for Accurate Matrix Multiplication

Check dimensions before calculation.
Use consistent decimal precision.
Remember: generally, A × B ≠ B × A.
Start with small test values to validate your setup.

Example

Suppose A is a 2×3 matrix and B is a 3×2 matrix. The result will be a 2×2 matrix. This calculator automatically builds the right input grids and computes the result instantly after you click Multiply A × B.

Final Notes

Use this tool as a fast checker while learning matrix algebra. Once you understand the row-by-column logic, you can confidently apply matrix multiplication across statistics, economics, optimization, and many technical fields.