Answer with Step-by-step explanation:
We are given that [tex]\lambda[/tex] is an eigenvalue of A.
We have to prove that [tex]\lambda^2[/tex] is an eigenvalue of [tex]A^2[/tex]
[tex]\lambda[/tex] is eigenvalue of A then, there exist an eigen vector x such that
[tex]Ax=\lambda x[/tex]
Multiply by A on both sides then we get
[tex]A(Ax)=A(\lambda x)[/tex]
[tex]A^2x=\lambda(Ax)[/tex]
[tex]A^2x=\lambda(\lambda x)[/tex]
[tex]A^2x=\lambda^2x[/tex]
Therefore, [tex]\lambda^2[/tex] is an eigen value of [tex]A^2[/tex]
Hence, proved.
