Matrix Multiplication Calculator

Enter the values for Matrix A and Matrix B (up to 3×3) and the Matrix Multiplication Calculator computes their product matrix C = A × B. Set the number of rows and columns for each matrix, fill in the cell values, and get the resulting matrix with all computed entries displayed below. Also try the Inverse Matrix Calculator.

Must equal Matrix A columns for multiplication to be valid.

Results

C[1][1]

--

C[1][2]

--

C[1][3]

--

C[2][1]

--

C[2][2]

--

C[2][3]

--

C[3][1]

--

C[3][2]

--

C[3][3]

--

Result Matrix Dimensions

--

Multiplication Valid

--

Results Table

Frequently Asked Questions

What is the main condition for multiplying two matrices?

For matrix multiplication A × B to be defined, the number of columns in Matrix A must equal the number of rows in Matrix B. If A is m×k and B is k×n, the result C will be an m×n matrix. If this condition is not met, the multiplication is undefined. See also our use the Matrix Determinant Calculator.

What size is the resulting matrix after multiplication?

The product matrix C = A × B has the same number of rows as Matrix A and the same number of columns as Matrix B. For example, multiplying a 2×3 matrix by a 3×2 matrix gives a 2×2 result matrix.

Is matrix multiplication commutative — does A × B equal B × A?

No, matrix multiplication is generally not commutative. In most cases A × B ≠ B × A. The dimensions may not even allow the reverse multiplication, and even when they do, the resulting values typically differ.

How is each entry of the product matrix calculated?

Each entry C[i][j] is calculated by taking the dot product of the i-th row of Matrix A and the j-th column of Matrix B. You multiply corresponding elements and sum the results: C[i][j] = Σ A[i][k] × B[k][j]. You might also find our Linear Independence useful.

Can I multiply non-square matrices?

Yes, you can multiply non-square matrices as long as the number of columns in the first matrix equals the number of rows in the second. For example, a 2×3 matrix can be multiplied by a 3×1 matrix to produce a 2×1 matrix.

What is an identity matrix and what happens when you multiply by it?

An identity matrix is a square matrix with 1s on the main diagonal and 0s elsewhere. Multiplying any matrix by a compatible identity matrix returns the original matrix unchanged — it acts like multiplying by 1 in scalar arithmetic.

Can this calculator handle negative numbers and decimals?

Yes, all matrix cell inputs accept negative numbers and decimal values. Enter them directly into the cell fields — for example, -3.5 or 0.25 — and the calculator will compute the correct product.

What does it mean if the calculator shows multiplication is not valid?

If the number of columns in Matrix A does not equal the number of rows in Matrix B, multiplication is mathematically undefined. Adjust your matrix dimensions so that A columns = B rows before proceeding.