Generated on: June 13, 2026 at 06:01 PM

Matrix Multiplication Calculator (2×2 and 3×3)

Formula

A = \begin{pmatrix} {{a11}} & {{a12}} \\ {{a21}} & {{a22}} \end{pmatrix}

B = \begin{pmatrix} {{b11}} & {{b12}} \\ {{b21}} & {{b22}} \end{pmatrix}

A = \begin{pmatrix} {{a11}} & {{a12}} & {{a13}} \\ {{a21}} & {{a22}} & {{a23}} \\ {{a31}} & {{a32}} & {{a33}} \end{pmatrix}

B = \begin{pmatrix} {{b11}} & {{b12}} & {{b13}} \\ {{b21}} & {{b22}} & {{b23}} \\ {{b31}} & {{b32}} & {{b33}} \end{pmatrix}

A \cdot B = \begin{pmatrix} {{c11}} & {{c12}} \\ {{c21}} & {{c22}} \end{pmatrix}

A \cdot B = \begin{pmatrix} {{c11}} & {{c12}} & {{c13}} \\ {{c21}} & {{c22}} & {{c23}} \\ {{c31}} & {{c32}} & {{c33}} \end{pmatrix}

Inputs

Matrix Size	2×2
A row 1, col 1	1
A row 1, col 2	2
A row 1, col 3	0
A row 2, col 1	3
A row 2, col 2	4
A row 2, col 3	0
A row 3, col 1	0
A row 3, col 2	0
A row 3, col 3	1
B row 1, col 1	5
B row 1, col 2	6
B row 1, col 3	0
B row 2, col 1	7
B row 2, col 2	8
B row 2, col 3	0
B row 3, col 1	0
B row 3, col 2	0
B row 3, col 3	1

Matrix Multiplication Calculator (2×2 and 3×3)

Multiply two square matrices instantly. Enter entries for 2×2 or 3×3 matrices A and B to compute the product A·B, with a live matrix visualization.

Matrix Size

2×2 3×3

A = \begin{pmatrix} {{a11}} & {{a12}} \\ {{a21}} & {{a22}} \end{pmatrix}

B = \begin{pmatrix} {{b11}} & {{b12}} \\ {{b21}} & {{b22}} \end{pmatrix}

Enter a value to see results.

Product A·B

A \cdot B = \begin{pmatrix} {{c11}} & {{c12}} \\ {{c21}} & {{c22}} \end{pmatrix}

Last updated: 2026-06-05

What this calculator does

This calculator multiplies two square matrices, A and B, to produce their product C = A·B. Choose 2×2 or 3×3 mode, enter the entries, and the product matrix updates instantly.

The entry in row i, column j of the result is the dot product of row i from A with column j from B:

c_{ij} = \sum_{k} a_{ik} \, b_{kj}

How matrix multiplication works

Each entry of the product is computed by pairing an entire row of A with an entire column of B:

C = A \cdot B = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix} = \begin{pmatrix} a_{11}b_{11}+a_{12}b_{21} & a_{11}b_{12}+a_{12}b_{22} \\ a_{21}b_{11}+a_{22}b_{21} & a_{21}b_{12}+a_{22}b_{22} \end{pmatrix}

For two n×n matrices, each entry requires n multiplications and (n−1) additions, so the total operation count is O(n³).

Worked example (2×2)

Let A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]]:

Entry (1,1): row 1 of A dot column 1 of B:

$c_{11} = (1)(5) + (2)(7) = 5 + 14 = 19$

Entry (1,2): row 1 of A dot column 2 of B:

$c_{12} = (1)(6) + (2)(8) = 6 + 16 = 22$

Entry (2,1): row 2 of A dot column 1 of B:

$c_{21} = (3)(5) + (4)(7) = 15 + 28 = 43$

Entry (2,2): row 2 of A dot column 2 of B:

$c_{22} = (3)(6) + (4)(8) = 18 + 32 = 50$

So A·B = [[19, 22], [43, 50]].

Non-commutativity

Matrix multiplication is not commutative. Swapping the order of the same two matrices gives:

B \cdot A = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = \begin{pmatrix} 23 & 34 \\ 31 & 46 \end{pmatrix}

A·B = [[19,22],[43,50]] but B·A = [[23,34],[31,46]]. These are different matrices. This is fundamentally unlike scalar multiplication, where 3×5 = 5×3 always.

The reason is geometric: A·B means "apply B first, then A," while B·A means "apply A first, then B." Different sequences of transformations generally produce different results.

Geometric interpretation

A square matrix encodes a linear transformation: it can rotate, scale, reflect, or shear a set of vectors. When you multiply A·B, you are composing two transformations — first B, then A. The result is a single matrix that performs both steps in one shot.

This composition rule is why matrix multiplication is associative (A·(B·C) = (A·B)·C) but not commutative.

The identity matrix

The n×n identity matrix I has 1s on the diagonal and 0s everywhere else. It plays the role of "multiply by 1" in matrix arithmetic:

$A \cdot I = I \cdot A = A$

Multiplying any matrix by the identity matrix leaves it unchanged. In geometric terms, I represents the transformation that does nothing — no rotation, no scaling.

Applications

Matrix multiplication appears whenever multiple linear transformations are chained:

3D graphics: Rotation, scaling, and translation are each 4×4 matrices (in homogeneous coordinates). Composing them by multiplication lets a GPU apply all transformations in a single matrix–vector product per vertex.
Systems of equations: The system Ax = b can be solved by computing A⁻¹b, where A⁻¹ is found by inverting the coefficient matrix.
Markov chains: Multiplying a transition matrix by itself k times gives the distribution after k steps.
Neural networks: Each layer of a fully connected network applies an affine transformation — a matrix multiplication followed by a bias addition.

Frequently Asked Questions (FAQ)

Is matrix multiplication commutative?

No. In general, A·B ≠ B·A. For example, with A = [[1,2],[3,4]] and B = [[5,6],[7,8]], A·B = [[19,22],[43,50]] but B·A = [[23,34],[31,46]]. Special cases like the identity matrix or diagonal matrices can commute, but this is the exception rather than the rule.

What does matrix multiplication mean geometrically?

Each matrix represents a linear transformation of space — a combination of rotations, reflections, scalings, and shears. Multiplying A·B computes the composition of those transformations: first apply B, then apply A. The order matters because applying a rotation then a shear gives a different result than applying the shear first.

Can I multiply non-square matrices?

Yes, as long as the number of columns in A equals the number of rows in B. An m×n matrix times an n×p matrix produces an m×p result. This calculator focuses on square 2×2 and 3×3 matrices, which are the most common cases in linear algebra courses and 2D/3D graphics.

How is matrix multiplication used in 3D graphics?

In 3D graphics, every rotation, scaling, and translation of an object is encoded as a 4×4 matrix (using homogeneous coordinates). To move an object through a sequence of transformations — say, rotate it, then scale it, then translate it — you multiply those three matrices together once to get a single combined matrix. That matrix is then applied to every vertex. This is why GPUs are optimized for matrix multiplication.

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