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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If a matrix A $\displaystyle \neq$ 0 and $\displaystyle A^k=0$ where k is a positive integer, does that mean A is singular?
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.
so the =0 means a matrix full of zeros right?
Yes, that is the 0 matrix.Quote:
Originally Posted by superdude
