The product of an m × p matrix A and a p × n matrix
B is an m × n matrix C where each element
c_{ij} is the dot product of row i of A and
column j of B.

$\mathbf{C=AB}$ | (12) |

implies

$c_{ij}=\displaystyle\sum_{k=1}^{p}(a_{ik}+b_{kj}),\operatorname{where}1\leq i% \leq m\operatorname{and}1\leq j\leq n$ | (13) |

The product of matrices A and B is undefined unless the number of rows in A is equal to the number of columns in B. In this case, the matrices are conformable for multiplication.

In general, matrix multiplication is not commutative.

$\mathbf{AB\neq BA}$ |

As a consequence, the following terminology is sometimes used. Considering the matrix product

$\mathbf{AB}$ |

The left multiplicand A is said to premultiply the matrix B. The right multiplicand B is said to postmultiply the matrix A.

Matrix multiplication distributes over matrix addition

$\mathbf{A(B+C)=\left(AB\right)+\left(AC\right)}$ |

and

$\mathbf{(B+C)A=\left(BA\right)+\left(CA\right)}$ |

if A, B, and C are conformable for the indicated operations. With the same caveat, matrix multiplication is associative.

$\mathbf{A\left(BC\right)=\left(AB\right)C}$ |

The transpose of a matrix product is the product of the factors in reverse order, i.e.

$\mathbf{{\left(ABC\right)}^{T}={C}^{T}{B}^{T}{A}^{T}}$ | (14) |

The set of square matrices has a multiplicative identity which is denoted by I. The identity is a diagonal matrix with ones along the diagonal

${a}_{ij}=\begin{cases}&1\operatorname{where}i=j\\ &0\operatorname{where}i\neq j\end{cases}$ | (15) |

The 3 × 3 multiplicative identity is

$\mathbf{I}=\left(\begin{array}[]{ccc}1&0&0\\ 0&1&0\\ 0&0&1\end{array}\right)$ |