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

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

implies

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

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

The 3 × 3 multiplicative identity is

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