does this mean a matrix is singular?

**superdude** · January 27th 2010, 08:32 PM

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

**nehme007** · January 27th 2010, 09:05 PM

A must be singular. Suppose not, then there exists $A^{-1}$ satisfying $AA^{-1}=A^{-1}A = I$ . Then $A^k (A^{-1})^k = I = 0$ , giving a contradiction.

**superdude** · January 27th 2010, 10:15 PM

so the =0 means a matrix full of zeros right?

**HallsofIvy** · January 27th 2010, 11:42 PM

Originally Posted by superdude

so the =0 means a matrix full of zeros right?

Yes, that is the 0 matrix.

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