If 𝜆 is an eigenvalue for 𝐴 and 𝐴3=0 show that 𝜆=0

Answer 1

Since λ is en eigenvalue of the matrix A we have that there exists a nonzero vector x such that Ax=λx. But then we simply have that 0=A^{3}x=A^{2}Ax=A^{2}(λx)=λA(Ax)=λ^{2}Ax=λ^{3}x and since x is a nonzero vector we have that λ^{3}=0 which is the same as saying that λ=0. We are done.