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2x2 Matrix Multiplication Calculator

Enter the entries of two 2×2 matrices to multiply them.

How it is calculated

The first matrix is A with entries a1, b1 on the top row and c1, d1 on the bottom row. The second is B with a2, b2 on top and c2, d2 below. Each entry of the product comes from a row of A and a column of B: r11 = a1·a2 + b1·c2, r12 = a1·b2 + b1·d2, r21 = c1·a2 + d1·c2, r22 = c1·b2 + d1·d2. The answer is shown as two rows, each with two numbers separated by a comma. All eight fields must be filled in; zero is allowed, a blank field is not. Negative numbers and decimals work, and values are shown with up to four decimals. The result is A × B, in that order. Swapping the two matrices usually gives a different answer, so check that A is in the first group of fields.

Worked example

Let A = [1 2; 3 4] and B = [5 6; 7 8]. Top-left: 1·5 + 2·7 = 5 + 14 = 19. Top-right: 1·6 + 2·8 = 6 + 16 = 22. Bottom-left: 3·5 + 4·7 = 15 + 28 = 43. Bottom-right: 3·6 + 4·8 = 18 + 32 = 50. The product A × B has rows 19, 22 and 43, 50. For comparison, B × A would be 23, 34 and 31, 46, so the order matters.

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Common mistakes

  • Multiplying entry by entry, so a1 with a2 and b1 with b2. Matrix multiplication pairs rows of the first matrix with columns of the second, which gives 19 and not 5 in the top-left of the example.
  • Assuming A × B equals B × A. They differ in general, and the tool only returns A × B for the order you entered.
  • Putting the entries in the wrong positions. The fields run row by row: a, b on top and c, d below, so c1 is the bottom-left entry of A, not the top-right.
  • Leaving a field empty when the entry is zero. Every one of the eight fields needs a number, and a blank gives no result.
  • Reading the output as a single list. Row 1 and row 2 are the two rows of the product matrix, not four separate answers.

About this calculator

Multiplying two 2×2 matrices follows the row-by-column rule: each entry of the result is the sum of products formed by pairing a row from the first matrix with a column from the second. Unlike ordinary number multiplication, matrix multiplication is not commutative — A×B generally gives a different result than B×A — because the rule pairs rows against columns in a fixed order that depends on which matrix comes first.

This operation is the mechanism behind composing linear transformations: multiplying a 2×2 rotation matrix by a scaling matrix, for example, produces a single matrix that rotates and scales in one step, which is exactly how 2D computer graphics, robotics kinematics and physics problems chain multiple transformations together efficiently. Small matrices like the 2×2 Pauli matrices also appear directly in quantum mechanics to represent spin operators.

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