Products of Real Matrices with Zero Matrices, Negatives and Differences

In the setting of Euclidean Space and Real Matrices: Standing Notation, let $p,q,r\in\mathbb{N}$, let $A,A'\in\mathcal{M}_{p\times q}(\mathbb{R})$ and $B,B'\in\mathcal{M}_{q\times r}(\mathbb{R})$, and let $x,y\in\mathbb{R}^{q}$.

(i) $A0_{q\times r}=0_{p\times r}$ and $0_{r\times p}A=0_{r\times q}$; $A0=0$, where $0$ on the left is the origin of $\mathbb{R}^{q}$ and on the right the origin of $\mathbb{R}^{p}$; and $0_{p\times q}x=0$, the origin of $\mathbb{R}^{p}$.

(ii) $(-A)B=-(AB)=A(-B)$ and $(-A)x=-(Ax)=A(-x)$.

(iii) $A(B-B')=AB-AB'$, $(A-A')B=AB-A'B$, $A(x-y)=Ax-Ay$ and $(A-A')x=Ax-A'x$.