Let A in M_m(F), be a Nilpotent matrix.

Prove the following:

1. A is a non-invertible matrix.

2. Order of nilpotentence of A small or equals to m.

3. I_n + A + A^2 +... + A^(k-1) is invertible when k is nilpotentence order of A.

4. If B is in M_n(F), a matrix. Is A+B must be Nilpotent matrix?

5. If B is in M_n(F), a matrix. Is A*B must be Nilpotent matrix?

6. Find B in M_5(R) Nilpotent matrix with order of 5.