I was doing linear algebra problems and got stuck in the following question:

Let $A$ and $B$ be $n\times n$ matrices such that $A^2 B^2=0$. Then which of the following option is correct? (and why?)

(a) $B^2 A^2=0$

(b) $AB=0$

(c) Either $A^2=0$ or $B^2=0$

(d) Either $A$ or $B$ has $0$ determinant

It would be helpful if any of you could provide a counterexample for the incorrect options. Thanks...

  • 2
    $\begingroup$ What are your thoughts? $\endgroup$ – Christoph Feb 3 '15 at 8:56
  • 1
    $\begingroup$ Hint: Either $\;A^2\;$ or $\;B^2\;$ is singular. What does this tell you about their determinants? $\endgroup$ – Timbuc Feb 3 '15 at 9:08
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    $\begingroup$ Use $\det(AB)=\det(A)\det(B)$ $\endgroup$ – Samrat Mukhopadhyay Feb 3 '15 at 9:11

Hint. Consider $A=\pmatrix{0&1&0\\ 0&0&1\\ 0&0&0}$ and some appropriate diagonal matrix $B$.


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