Free Matrix Multiplication Calculator

Multiply Matrices with the Free Online Matrix Product Calculator

This matrix multiplication calculator is a free online tool that computes the product of two matrices (often denoted as AB). Handling sizes such as 2x2 matrix multiplication and 3x3 matrix multiplication, it quickly calculates every entry of the resulting matrix.

What Is Matrix Multiplication?

Unlike ordinary number multiplication, multiplying two matrices follows a specific rule based on the dot product of rows and columns. For matrices A (with $r$ rows and $s$ columns) and B (with $s$ rows and $t$ columns), the product $\mathbf{C} = \mathbf{A}\mathbf{B}$ exists only if the number of columns of A equals the number of rows of B. The resulting matrix C then has dimensions $r \times t$.

The entry in row $i$ and column $j$ of C is the sum of products of the corresponding elements from row $i$ of A and column $j$ of B:

$$c_{ij} = \sum_{k=1}^{s} a_{ik} b_{kj}$$

Each $c_{ij}$ is essentially the matrix dot product of the $i$-th row of A and the $j$-th column of B.

Key Rules and Properties

• Dimension prerequisite: The inner dimensions must match — if A is $m \times n$, B must be $n \times p$.
• Not commutative: In general $\mathbf{A}\mathbf{B} \neq \mathbf{B}\mathbf{A}$. Often one product is not even defined because the dimensions are incompatible.
• Generalizes scalar multiplication: When both matrices are $1 \times 1$, their product is just the product of the two numbers.
• Determinant property: For square matrices, $\det(\mathbf{A}\mathbf{B}) = \det(\mathbf{A})\det(\mathbf{B})$.

Step‑by‑Step Example: 3×2 Times 2×2

Let

$$\mathbf{A} = \begin{pmatrix} 3 & 2\\ 1 & 6\\ 5 & 4 \end{pmatrix},\qquad \mathbf{B} = \begin{pmatrix} 5 & 2\\ 7 & 1 \end{pmatrix}.$$

Since A has 3 rows and 2 columns and B has 2 rows and 2 columns, the product $\mathbf{A}\mathbf{B}$ is a $3 \times 2$ matrix. Each entry is computed as the dot product of a row from A and a column from B:

• $c_{11} = (3)(5) + (2)(7) = 15 + 14 = 29$
• $c_{12} = (3)(2) + (2)(1) = 6 + 2 = 8$
• $c_{21} = (1)(5) + (6)(7) = 5 + 42 = 47$
• $c_{22} = (1)(2) + (6)(1) = 2 + 6 = 8$
• $c_{31} = (5)(5) + (4)(7) = 25 + 28 = 53$
• $c_{32} = (5)(2) + (4)(1) = 10 + 4 = 14$

Thus

$$\mathbf{A}\mathbf{B} = \begin{pmatrix} 29 & 8\\ 47 & 8\\ 53 & 14 \end{pmatrix}.$$

This same process is automated inside the matrix product calculator, saving time and avoiding manual errors.

How to Use the Matrix Multiplication Calculator

1. Select the number of rows and columns for the first matrix (e.g., 3 rows, 2 columns).
2. Do the same for the second matrix (e.g., 2 rows, 2 columns). The calculator will only allow dimension pairs that are compatible for multiplication.
3. Enter the numerical values in the grid that appears.
4. Click the “Calculate” button to instantly obtain the AB matrix product.

You can use this tool for 2x2 matrix multiplication, 3x3 matrix multiplication, or any other combination where the inner dimensions agree. The result is displayed clearly, and you can also see the intermediate dot‑product steps if needed.

Going Further: Matrix Decompositions

Just as numbers can be factored, matrices can be decomposed into products of simpler matrices. Common techniques include LU decomposition and Cholesky decomposition, which are valuable in solving linear systems and other advanced applications. While this calculator focuses on direct matrix multiplication, understanding these factorisations deepens your grasp of matrix theory.

Whether you’re studying linear algebra, working with 3D graphics, or solving systems of equations, the ability to multiply matrices quickly and accurately is essential. This online matrix product calculator gives you a reliable, free way to perform the operation for any compatible pair of matrices.