Assume that matrix $A \in M_n$ is similar to a diagonal matrix $D$ with $0$ or $1$ diagonal entries. I read the textbook and it says that there are $n+1$ such different diagonal matrices. However, as I expect, if each entries can select values $0$ or $1$, we should have $(2!)^n$ such kind of matrices.
Could you please point out what is wrong in my thinking?
Note that, for example, the matrices $$ \pmatrix{1\\&0\\&&0}, \pmatrix{0\\&1\\&&0} $$ are similar.