V = Mnxn is a space vector formed by all the square matrices.

Question

Is W = {A in Mnxn : A^x = A} a subspace of V?

Thanks

Answer 1

No, because if A is in W, then

B = lambda A

is not in W if lambda is not equal to 1:

A^x = A --->

B^x = lambda^x A^x = lambda^x A =

labda^(x-1) B

which is not equal to B for general lambda.