Matrix Addition/Subtraction Properties

Question

I was wondering if anyone could explain to me how the following properties are derived for square matrices of the same size. I know that it is possible to prove/disprove the following properties by showing that they hold for the arbitrary elements of A and B, but is there a quicker way to determine whether these are true or false?

1. $(A−B)^2 =A^2 −2AB+B^2$

2. $(AB)^2 = A^2B^2$

3. $(A+B)^2 =A^2 +2AB+B^2$

4. $(A+B)^2 =A^2 +AB+BA+B^2$

5. $A^2B^2 =A(AB)B$

6. $(A+B)^3 =A^3 +3A^2B+3AB^2 +B^3 $

7. $ (A+B)(A−B)=A^2 −B^2$

Answer 1

Hint:

1. In general, $AB\ne BA$ (in other words, not commutative), applicable for all but 5
2. Matrix multiplication is distributive, applicable for all but 5
3. Matrix multiplication is associative, you can use it for 5.