Suppose that $M_{mn}$ is the set of all $m\times n$ matrices (Definition VSM) with addition and scalar multiplication as defined in Definition MA and Definition MSM. Then

• ACM Additive Closure, Matrices
  If $A,\,B\in M_{mn}$, then $A+B\in M_{mn}$.
  
  
• SCM Scalar Closure, Matrices
  If $\alpha\in\complex{\null}$ and $A\in M_{mn}$, then $\alpha A\in M_{mn}$.
  
  
• CM Commutativity, Matrices
  If $A,\,B\in M_{mn}$, then $A+B=B+A$.
  
  
• AAM Additive Associativity, Matrices
  If $A,\,B,\,C\in M_{mn}$, then $A+\left(B+C\right)=\left(A+B\right)+C$.
  
  
• ZM Zero Matrix, Matrices
  There is a matrix, $\zeromatrix$, called the zero matrix, such that $A+\zeromatrix=A$ for all $A\in M_{mn}$.
  
  
• AIM Additive Inverses, Matrices
  If $A\in M_{mn}$, then there exists a matrix $-A\in M_{mn}$ so that $A+(-A)=\zeromatrix$.
  
  
• SMAM Scalar Multiplication Associativity, Matrices
  If $\alpha,\,\beta\in\complex{\null}$ and $A\in M_{mn}$, then $\alpha(\beta A)=(\alpha\beta)A$.
  
  
• DMAM Distributivity across Matrix Addition, Matrices
  If $\alpha\in\complex{\null}$ and $A,\,B\in M_{mn}$, then $\alpha(A+B)=\alpha A+\alpha B$.
  
  
• DSAM Distributivity across Scalar Addition, Matrices
  If $\alpha,\,\beta\in\complex{\null}$ and $A\in M_{mn}$, then $(\alpha+\beta)A=\alpha A+\beta A$.
  
  
• OM One, Matrices
  If $A\in M_{mn}$, then $1A=A$.