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2x2 Matrix Multiplication Calculator

Multiply two 2×2 matrices A × B and get every entry of the product from the row-by-column dot products.

A — Row 1, Col 1

A — Row 1, Col 2

A — Row 2, Col 1

A — Row 2, Col 2

B — Row 1, Col 1

B — Row 1, Col 2

B — Row 2, Col 1

B — Row 2, Col 2

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How it works

AB[i][j] = row i of A · column j of B · (AB)₁₁ = ae + bg, (AB)₁₂ = af + bh, (AB)₂₁ = ce + dg, (AB)₂₂ = cf + dh

Each entry of the product is the dot product of a row of A with a column of B. The (i, j) entry pairs row i with column j, multiplying matching elements and summing. Matrix multiplication is not commutative — AB and BA are generally different matrices — so the order you enter A and B matters.

Worked example

With A = [1, 2; 3, 4] and B = [5, 6; 7, 8]: (AB)₁₁ = 1·5 + 2·7 = 19, (AB)₁₂ = 1·6 + 2·8 = 22, (AB)₂₁ = 3·5 + 4·7 = 43, (AB)₂₂ = 3·6 + 4·8 = 50, giving [19, 22; 43, 50].

Sources: Matrix multiplication (Wikipedia)

Frequently asked questions

Is matrix multiplication commutative?

No. In general AB ≠ BA, even for 2×2 matrices. Swapping the order usually changes the result, which is why this calculator keeps A and B distinct.

How is each entry of the product formed?

The entry in row i, column j is the sum of the products of row i of A with column j of B, element by element. For 2×2 matrices that is two products added together per entry.

What is the identity matrix here?

The 2×2 identity is [1, 0; 0, 1]. Multiplying any matrix by it (on either side) leaves that matrix unchanged, the matrix equivalent of multiplying a number by 1.

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This calculator is for educational purposes. Double-check important results before acting on them.