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2×2 Matrix Multiplication Calculator

Multiply two 2×2 matrices.

How it is calculated

The first matrix is A = [[a, b], [c, d]] (first row a, b; second row c, d) and the second is B = [[e, f], [g, h]]. The product A × B has four entries:

r11 = a·e + b·g, r12 = a·f + b·h, r21 = c·e + d·g, r22 = c·f + d·h.

Each entry takes a row of A and a column of B, multiplies matching entries and adds. All eight fields must be filled; negatives and decimals are fine, and the results are shown with two decimals. The order matters: the first matrix always goes on the left, so swapping A and B generally changes the result.

Worked example

Let A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]]. Then r11 = 1×5 + 2×7 = 19.00; r12 = 1×6 + 2×8 = 22.00; r21 = 3×5 + 4×7 = 43.00; r22 = 3×6 + 4×8 = 50.00. The product is [[19, 22], [43, 50]]. Reversing the order (B × A) gives [[23, 34], [31, 46]], a different matrix.

Try this example →

Common mistakes

  • Multiplying entry by matching entry (a·e, b·f, …). That is not matrix multiplication; rows are combined with columns.
  • Entering a matrix column by column, so b and c are swapped; the result then corresponds to the transpose.
  • Assuming A × B equals B × A. The two products usually differ, as in the example.
  • Mixing up the field letters: the second matrix is e, f in its top row and g, h in the bottom row.
  • Expecting the displayed two decimals to be the exact value when you enter long decimals; the display is rounded.

About this calculator

This calculator multiplies two 2×2 matrices using the standard row-by-column dot product rule: each entry of the result is the sum of products of a row from the first matrix and a column from the second, meaning matrix multiplication order matters (A×B usually isn't the same as B×A).

It's used by students learning linear algebra fundamentals, and in computer graphics and game development, where 2×2 matrices represent 2D transformations like rotation and scaling that get applied to points and vectors.

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Comments (1)

  • Editorial team

    Two quick checks help. Multiplying by the identity [[1, 0], [0, 1]] must return the other matrix unchanged. And order matters: with A = [[0, −1], [1, 0]] (a 90 degree turn) and B = [[1, 0], [0, 2]] (a stretch), A × B is [[0, −2], [1, 0]] but B × A is [[0, −1], [2, 0]]. What are you multiplying them for, transformations or something else?

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