If a matrix A $\displaystyle \neq$ 0 and $\displaystyle A^k=0$ where k is a positive integer, does that mean A is singular?

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- Jan 27th 2010, 08:32 PMsuperdudedoes this mean a matrix is singular?
If a matrix A $\displaystyle \neq$ 0 and $\displaystyle A^k=0$ where k is a positive integer, does that mean A is singular?

- Jan 27th 2010, 09:05 PMnehme007
A must be singular. Suppose not, then there exists $\displaystyle A^{-1}$ satisfying $\displaystyle AA^{-1}=A^{-1}A = I$. Then $\displaystyle A^k (A^{-1})^k = I = 0$, giving a contradiction.

- Jan 27th 2010, 10:15 PMsuperdude
so the =0 means a matrix full of zeros right?

- Jan 27th 2010, 11:42 PMHallsofIvy