I came across this problem which says:
Let $A$ be a $2 \times 2$ matrix such that only $A$ is similar to itself.Then show that A is a scalar matrix, that is $A$ is of the form \begin{pmatrix} a &0 \\ 0 & a \end{pmatrix} ? My attempts: Since $A$ is similar to itself,there exists an invertible matrix P such that A= $P^{-1}AP$. Then I tried to solve it by choosing A and P of the form \begin{pmatrix} a &b \\ c & d \end{pmatrix} and \begin{pmatrix} x &y \\ z & w \end{pmatrix} respectively. But I could not get the desired result. Please help. Thanks everyone in advance for your time.