Matrix multiplication, rules

Summary

• $A,B,C$ are matrices with dimensions such that the products below are defined. $α$ is a scalar.
• $AB$ is not necessarily equal to $BA$ (commutative law does not hold for matrix multiplication)
• $(AB)C=A(BC)$ (the associative law holds for matrix multiplication)
• $A(B+C)=AB+AC$ (the distributive law holds for matrices)
• $(A+B)C=AC+BC$
• $α(AB)=(αA)B=A(αB)$
• $AI=IA=A$ where $I$ is the identity matrix
• $(AB)^T=B^TA^T$