Let $A$ be a $2\times2$ real square matrix of rank $1$. If $A$ is not diagonalizable, then which of the following is true.
(a) $A$ is nilpotent
(b) $A$ is not nilpotent
(c) the characteristic polynomial of $A$ is linear.
(d) $A$ has a non-zero eigenvalue.
I can say that d is false.