# 2.5 Matrix Multiplication

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

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

implies

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

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