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Matrix Multiplication Calculator

Multiply matrices up to 4×4 and see every row-by-column dot product. Free matrix calculator for linear algebra, A-level Further Maths and JEE.

Matrix A (2×2)
Rows A2
Cols A2
Matrix B (2×2)
Rows B2
Cols B2

A × B

(1234)(5678)=(19224350)\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}
  1. 1

    Check the sizes

    A is 2×2 and B is 2×2. The inner sizes match (2 = 2), so AB exists and is 2×2.

    A=(1234),B=(5678)A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}
  2. 2

    c11: row 1 of A · column 1 of B

    c11=1⋅5+2⋅7=5+14=19c_{11} = 1\cdot 5 + 2\cdot 7 = 5 + 14 = 19
  3. 3

    c12: row 1 of A · column 2 of B

    c12=1⋅6+2⋅8=6+16=22c_{12} = 1\cdot 6 + 2\cdot 8 = 6 + 16 = 22
  4. 4

    c21: row 2 of A · column 1 of B

    c21=3⋅5+4⋅7=15+28=43c_{21} = 3\cdot 5 + 4\cdot 7 = 15 + 28 = 43
  5. 5

    c22: row 2 of A · column 2 of B

    c22=3⋅6+4⋅8=18+32=50c_{22} = 3\cdot 6 + 4\cdot 8 = 18 + 32 = 50
  6. 6

    Assemble the result

    Place each cᵢⱼ in row i, column j.

    AB=(19224350)AB = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}

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FAQ

When can two matrices be multiplied?

A (m×n) times B (n×p) works only when the columns of A equal the rows of B; the result is m×p.

Is matrix multiplication commutative?

No — in general AB ≠ BA.

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