Matrix Multiplication Walkthrough

Matrix multiplication isn't entry-by-entry like addition — it's a row-by-column operation. Once you see the pattern it's mechanical, and a matrix multiplication calculator confirms every answer. Let's build it up.

The dimension rule

You can multiply an m×n matrix by an n×p matrix — the inner numbers must match — and the result is m×p:

(m × n) · (n × p)  →  (m × p)
        ↑↑↑↑↑ must be equal

So a 2×3 times a 3×2 works and gives a 2×2; a 2×3 times a 2×2 does not work at all.

One entry = one dot product

Entry (i, j) of the product is the dot product of row i of the first matrix with column j of the second: multiply matching elements and add.

Full worked example

Multiply A × B where

A = 1 2     B = 5 6
    3 4         7 8

Compute each of the four entries:

• (1,1) = row 1 · col 1 = 1·5 + 2·7 = 5 + 14 = 19
• (1,2) = row 1 · col 2 = 1·6 + 2·8 = 6 + 16 = 22
• (2,1) = row 2 · col 1 = 3·5 + 4·7 = 15 + 28 = 43
• (2,2) = row 2 · col 2 = 3·6 + 4·8 = 18 + 32 = 50

Order matters

Reverse the order and you get a different matrix:

Check on the calculator

In MATRIX mode, enter A in the first box and B in the second:

Matrix A      Matrix B
1 2           5 6
3 4           7 8

The calculator returns A × B = [[19, 22], [43, 50]] (plus A + B). Swap the inputs to see B × A change. For determinants, inverses and transposes of the same matrices, see the matrix calculator pillar guide, and to apply products to systems read simultaneous equations with matrices.

Frequently asked questions