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Matrix Multiplication Calculator

Multiply two 3x3 matrices with the row-by-column dot product rule. Editable grids, live results.

Matrix Multiplication

Fill both 3x3 grids. Results update live.

Result
A x B

Edit the grids to compute.

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Rule: C[i][j] = sum over k of A[i][k] x B[k][j]. Note that A x B is generally not equal to B x A.
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Rows times columns

Each entry of the product is a dot product: take row i of A and column j of B, multiply the pairs, and add. For 3x3 matrices that is three multiplications per entry, 27 in total, which the tool recomputes on every keystroke.

Order matters

Matrix multiplication is not commutative: A x B usually differs from B x A. Swapping the factors answers a different question, so double check which matrix goes on the left before reading the result.

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Matrix Multiplication FAQs

What is [[1,2],[3,4]] times [[5,6],[7,8]]?
Enter them into the top-left 2x2 of the grids (rest zeros): the product is [[19,22],[43,50]]. Each entry is a row-by-column dot product.
Is matrix multiplication commutative?
No. In general A x B does not equal B x A, so the order of the two grids matters. Try swapping two non-trivial matrices to see.
What sizes can be multiplied?
An m x n matrix times an n x p matrix gives m x p. This tool multiplies 3x3 by 3x3, which always works.
What is the identity matrix?
The 3x3 identity has 1s on the diagonal and 0s elsewhere: [[1,0,0],[0,1,0],[0,0,1]]. Multiplying any matrix by it leaves the matrix unchanged.
Are my matrices uploaded anywhere?
No. All 27 multiplications run in your browser; your data never leaves the page.
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