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.

C = AB

implies

cij=k=1p(aik+bkj),where1imand1jnc_{ij}=\displaystyle\sum_{k=1}^{p}(a_{ik}+b_{kj}),\operatorname{where}1\leq i% \leq m\operatorname{and}1\leq j\leq n (3)

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.

AB ≠ BA

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

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

A(B + C) = AB + AC

and

(B + C)A = BA + CA

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

A(BC) = (AB)C

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

(ABC)T = CTBTAT

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

aij={1wherei=j0whereij{a}_{ij}=\begin{cases}&1\operatorname{where}i=j\\ &0\operatorname{where}i\neq j\end{cases} (4)

The 3 × 3 multiplicative identity is

𝐈=(100010001)\mathbf{I}=\left(\begin{array}[]{ccc}1&0&0\\ 0&1&0\\ 0&0&1\end{array}\right)