# 2013-14 Nilpotent matrix

Let $$A, B$$ are $$N \times N$$ complex matrices satisfying $$rank(AB – BA) = 1$$. Prove that $$(AB – BA)^2 = 0$$.

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# 2010-10 Metric space of matrices

Let  Mn×n be the space of real n×n matrices, regarded as a metric space with the distance function

$$\displaystyle d(A,B)=\sum_{i,j} |a_{ij}-b_{ij}|$$

for A=(aij) and B=(bij).
Prove that $$\{A\in M_{n\times n}: A^m=0 \text{ for some positive integer }m\}$$ is a closed set.

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