Let A be an n x n matrix such that A^3 = On

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I have a problem.
Let A be an n x n matrix such that A^3 = On (On is a n x n zero matrix). Show that the only possible eigenvalue for A is 0.
 

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jbunniii
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An eigenvalue lambda must satisfy

Ax = lambda x

for some nonzero vector x.

Now multiply both sides of the above equation on the left by A, twice, and substitute lambda x in place of Ax where appropriate.

What is the resulting equation, and what does it imply about lambda?
 

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