Thread: Nilpotent matrix Invertible matrix

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.

Originally Posted by 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.

.

Yes! it's my mistake. A is a non-invertible matrix.