Chapter A.1, Problem 4P

Prove each of the following laws of matrix algebra:

(a) $\text{A+B}\text{=}\text{B+A}$

(b) $\text{A+}\left(\text{B+C}\right)\text{=}\left(\text{A+B}\right)\text{+C}$

(c) $\text{α}\left(\text{A+B}\right)\text{=αA+αB}$

(d) $\left(\alpha +\beta \right)\text{A=}\alpha \text{A+}\beta \text{A}$

(e) $\text{A}\left(\text{BC}\right)\text{=}\left(\text{AB}\right)\text{C}$

(f) $\text{A}\left(\text{B+C}\right)\text{=AB+AC}$