# does this mean a matrix is singular?

• January 27th 2010, 09:32 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, 10:05 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, 11:15 PM
superdude
so the =0 means a matrix full of zeros right?
• January 28th 2010, 12:42 AM
HallsofIvy
Quote:

Originally Posted by superdude
so the =0 means a matrix full of zeros right?

Yes, that is the 0 matrix.