Matrix multiplication

Multiplying matrices is done row times column, and is not the same as multiplying entrywise. The order matters: in general $AB\neq BA$ .

(AB)_{ij} = \sum_k A_{ik}B_{kj}

entry

ij

of the product, row i of A times column j of B

(AB)^T = B^TA^T

transpose of a product (order is reversed)

Symbols

A^T

transpose of A (rows and columns swapped)

$A=\begin{pmatrix}-3&-2\\3&-3\end{pmatrix}$ , $B=\begin{pmatrix}-2&-2\\-1&0\end{pmatrix}$ : $(AB)_{11}=(-3)(-2)+(-2)(-1)=8$ .

AB\neq BA

in general - the order of the matrices matters, just like the order of rotations.