does this mean a matrix is singular?

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January 27th 2010, 08:32 PM
superdude

does this mean a matrix is singular?

If a matrix A $\neq$ 0 and $A^k=0$ where k is a positive integer, does that mean A is singular?
January 27th 2010, 09:05 PM
nehme007

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.
January 27th 2010, 10:15 PM
superdude

so the =0 means a matrix full of zeros right?
January 27th 2010, 11:42 PM
HallsofIvy

Quote:

Originally Posted by superdude

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

Yes, that is the 0 matrix.

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