Which of the statements $($a$)($d$)$ is false?

Let A and B be $12\backslash$times $12$ matrices, and let v in R$^12$ be a common eigenvector for

A and B$.$ Then

$($a$)$ A v is an eigenvector of A;

$($b$)$ v is an eigenvector for A$^2$;

$($c$)$ v is an eigenvector for A $+$ B;

$($d$)$ v is an eigenvector for A B;

$($e$)$ none of the above $($statements $($a$)($d$)$ are true for all matrices A$,$ B and all

eigenvectors v in R

$12$ as above$).$