Exercise 4

Prove each, assuming that the operations are defined, where $G$, $H$, and $J$ are matrices, where $Z$ is the zero matrix, and where $r$ and $s$ are scalars.

• Matrix addition is commutative $G+H=H+G$.

• Matrix addition is associative $G+(H+J)=(G+H)+J$.

• The zero matrix is an additive identity $G+Z=G$.

• $0\cdot G=Z$

• $(r+s)G=rG+sG$

• Matrices have an additive inverse $G+(-1)\cdot G=Z$.

• $r(G+H)=rG+rH$

• $(rs)G=r(sG)$