To multiply two matrices, take each row of the left matrix and dot it with each column of the right matrix. The entry in row i, column j of the product is

For this to work the rows of A and columns of B must be the same length — so the columns of A must match the rows of B. An (m\times n) times an (n\times p) gives an (m\times p). The shared n is consumed.

Row dot column, cell by cell

Step through the four output cells. Each one lights up a row of A and a column of B, and their dot product fills the answer. Four little dot products build the whole product.

Why "row dot column"?

It looks arbitrary until you remember what a matrix does. Multiplying by B then by A means transforming a vector twice: A(B\vec{x}). The product matrix AB is the single matrix that does both steps at once — which is exactly why composing transformations is matrix multiplication, and why a deep neural network is just a chain of matrix products.