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

[squenshl]

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.

 

[jbunniii]

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?