1.4.1—Matrix operations

To multiply A by B, the number of columns of A must equal the number of rows of B. Each entry of AB is the dot product of a row of A with a column of B.

The order matters: AB may exist while BA does not, or both may exist with different results. Matrix addition requires equal dimensions, while scalar multiplication scales every entry.

For A=[[1,2],[3,4]] and B=[[0,1],[2,0]], AB=[[4,1],[8,3]], whereas BA=[[3,4],[1,2]].

Matrix multiplication is not entry-by-entry multiplication, and AB=BA is a special property rather than a default.