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**Also sprach Zarathustra** Let A in M_m(F), be a Nilpotent matrix.

Prove the following:

**1. A is a invertible matrix.**

** This is false, of course: if a matrix mis nilpotent then its determinant is zero and thus it isn't invertible.**

**Tonio**

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.