$A$ be a $n\times n$ real matrix, its diagonalizability with specific conditions

Question

$A$ be a $n\times n$ real matrix, could any one tell me which of the following is correct?

1. If $A^2=0$, then $A$ is diagonalizable over $\mathbb{C}$.

2. If $A^2=I$, then $A$ is diagnalizble over reals.

3. If $A$ is idempotent then it is diagnalzble only over complex numbers.

4. the only matrix of size $n$ satisfying the char.poly of $A$ is $A$.

$1$ is not true ingeneral, in this case minpoly must divide $x^2$, so minpoly may be $x,x^2$, if minpoly is $x=0$ the we see $A=0$, and if $x^2=0$ then not diagonalizble as it has no distinct roots.

I have not clear idea about $2$, in this case minpoly may be $x^2-1=0,x+1=0,x-1=0$ so I guess it is diagonalizable due to distinct roots, $A=I,-I$

in case $3$ only eigen values are $0,1$ but I have no idea bout multiplicity, minpoly will be $x,x(x-1),(x-1)$, I can not say anything here.

$4$ is definitely false.

Thank you for helping.

Answer 1

1. No, for example $$ A=\left(\begin{matrix}0 & 1 \\ 0 & 0\end{matrix}\right) $$

2. True. If a matrix is annihilated by a polynomial with single roots, then it is diagonalizable, and its eigenvalues are roots of the polynomial (here $x^2-1$.)

3. They are diagonalizable as the polynomial $x^n-1$ has only single roots. But, not necessarily over complex numbers, i.e., $$ A=\left(\begin{matrix}1 & 0 \\ 0 & -1\end{matrix}\right) $$

4. Not true. $$ A=\left(\begin{matrix}0 & 1 \\ 0 & 0\end{matrix}\right) \quad\text{and}\quad B=\left(\begin{matrix}0 & 0 \\ 0 & 0\end{matrix}\right) $$ satisfy $p(x)=x^2$, which is their char. poly.