Let $A, B, C \in \mathrm{Mat}_{m n}$ and let $k, s \in \mathbb{R}$ be scalars. The following identities hold: (i) $A+0_{m n}=A$ (ii) $A+B=B+A$ (iii) $0 A=0_{m n}$ (iv) $A+(-A)=0_{m n}$ (v) $(A+B)+C=A+(B+C)$ (vi) $1 A=A$ (vii) $(k+s) A=k A+s A$ (viii) $k(A+B)=k A+k B$ (ix) $k(s A)=(k s) A$