Matrix Multiplication

Suppose $A=(a_{ij})_{m\times{p}}$ and $B=(b_{ij})_{q\times{n}}$. Matrix multiplication is only defined when $q=p$ here.

$$A\times{B}=C =(c_{ij})_{m\times{n}} \;\;\text{where} \;\; c_{ij} = \sum_{k=1}^{p}{a_{ik}\times b_{kj}}$$

Note

• Generally $A\times{B}\not=B\times{A}$
• $A\times{B}=0 \centernot\implies A=0 \lor B=0$
• $A\not=0 \land B\not=0 \centernot\implies A\times{B}\not=0$

Properties of matrix multiplication

$A,B,C,I$ matrices must be chosen so that below-mentioned products are defined. $$

1. Associative: $A(BC) = (AB)C$
2. Right distributive over addition: $(A+B)C=AC+BC$
3. Left distributive over addition: $C(A+B)=CA+CB$
4. $AI=IA=A$